6. influence lines for statically determinate structures

Department of Civil Engineering NPIC

^> ExST§iBlsMrab;rcnasm<½n§kMNt;edaysþaTic
Influence lines for statically determinate structures

ExST§iBlmansar³sMxan;kñúgkarGnuvtþsMrab;karsikSaKNnaeRKOgbgÁúMEdlTb;Tl;nwgbnÞúkGrefrEdl
mantMélFM. CMBUkenHmankarerobrab;BIrebobKUrExST§iBlsMrab;eRKOgbgÁúMkMNt;edaysþaTic. RTwsþI
RtUv)anGnuvtþeTAelIbnÞúkBRgay b¤es‘rIénbnÞúkcMcMnuc ehIyk¾pþl;nUvkarGnuvtþCak;lak;eTAelIr:tkMral
(floor girder) nig truss s<an. karKNnakMlaMgkat; nigm:Um:g;Gtibrmadac;xatenAkñúgGgát;RtUv)anerob

rab;enAcugbBa©b;énCMBUkenH.
^>!> ExST§iBl (Influence line)
eyIg)ansikSarYcehIyBIbec©keTssMrab;viPaKkMlaMgenAkñúgGgát;eRKOgbgÁúMEdlekItBIbnÞúkmin
cl½t ¬Kμanclna¦. eyIgeXIjfadüaRkamkMlaMgkat; nigdüaRkamm:Um:g;Bt;CameFüa)ayd¾RbesIrkñúgkar
bgðajBIbMErbMrYlkMlaMgTaMgenHenAkñúgGgát;. b:uEnþRbsinebIeRKOgbgÁúMrgbnÞúkcl½t ¬manclna¦ ExST§i-
Bl (influence line) CameFüa)ayd¾l¥bMputkñúgkarbgðajBIbMErbMrYlkMlaMgkat; nigm:Um:g;Bt;EdlekItman
enAkñúgGgát;eRKOgbgÁúM. ExST§iBlbgðajbMErbMrYlrbs;kMlaMgRbtikmμ kMlaMgkat; m:Um:g;Bt; nigPaBdab
enARtg;cMnucCak;lak;enAelIFñwmeRkamGMeBIénkarcl½trbs;bnÞúkcMcMnuc. enAeBlEdleKsg;ExST§iBlrYc
ehIy eKGacR)ab;BITItaMgrbs;bnÞúkcl½tedIm,IbegáIteGaymanT§iBlFMbMput. elIsBIenH eKGackMNt;
TMhMrbs;kMlaMgRbtikmμ kMlaMgkat; m:Um:g;Bt; b¤PaBdabeRkamGMeBIrbs;bnÞúkenARtg;cMnucNamYyBIGreda-
en rbs;düaRkamExST§iBl. sMrab;mUlehtuenH ExST§iBledIrtYy:agsMxan;kñúgkarsikSaKNnas<an/ pøÚv
rt;rbs;]bkrN_eyag (industrial crane rails, conveyor) nigeRKOgbgÁúMdéTeTotEdlbnÞúkcl½telI
ElVgrbs;va.
eTaHbICadMeNIrkarsagsg;ExST§iBlmanlkçN³mUldæank¾eday EteKRtUvEbgEckeGaydac;BI
Kñarvagkarsg;ExST§iBl nigkarsg;düaRkamkMlaMgkat; nigdüaRkamm:Um:g;Bt;. ExST§iBlbgðajEtBI
T§iBlrbs;bnÞúkcl½tenARtg;TItaMgCak;lak;enAelIGgát; EtdüaRkamkMlaMgkat; nigdüaRkamm:Um:g;bgðaj
BIT§iBlrbs;bnÞúkGcl½tenARKb;cMnucTaMgGs;tambeNþayG½kSrbs;Ggát;.

dMeNIrkarkñúgkarviPaK (Procedure for analysis)
xageRkamCaviFIBIrEdlRtUv)aneKeRbIedIm,Isg;ExST§iBlenARtg;cMnucCak;lak; P enAkñúgGgát;
ExST§iBlsMrab;rcnasm<½n§kMNt;edaysþaTic T.Chhay -168

mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa

sMrab;kMlaMgRbtikmμ kMlaMgkat; nigm:Um:g;Bt;. sMrab;viFITaMgBIrenHeyIgnwgeRCIserIskMlaMgcl½tEdlman
TMhMKμanxñatmYyÉktþa mkeRbIsMrab;sg;ExST§iBl.
*

CMnYstMélelxkñúgtarag (tabulate values)
dak;bnÞúkÉktþaenARtg;TItaMg ¬ x ¦epSg² tambeNþayGgát; ehIyenARtg;TItaMgnImYy²eRbI
sþaTicedIm,IkMNt;kMlaMgRbtikmμ kMlaMgkat; m:Um:g;Bt;enARtg;cMnucCak;lak;.
RbsinebIeKcg;sg;ExST§iBlsMrab;kMlaMgRbtikmμbBaÄrenARtg;cMnucmYyenAelIFñwm eKRtUvKitfa
kMlaMgRbtikmμmanTMhMviC¢manenAeBlvaeFVIGMeBIeLIgelI.
RbsinebIeKKUrExST§iBlkMlaMgkat; nigExST§iBlm:Um:g;Bt;sMrab;cMnucmYy ykkMlaMgkat; nig
m:Um:g;Bt;viC¢mandUckarkMNt;sBaØaEdleRbIsMrab;KUrdüaRkamkMlaMgkat; nigdüaRkamm:Um:g;Bt;
¬emIlrUbTI 4-1¦.
FñwmkMNt;edaysþaTicTaMgGs;manExST§iBlEdlpSMeLIgedaykMNat;bnÞat;Rtg;. eRkayBIkar
GnuvtþxøH eKGackat;bnßykarKNnaedayeGaybnÞúkÉktþamanGMeBIEtenAcMnuccugénkMNat;Ggát;
nImYy².
edIm,IeCosvagkMhusqÁg dMbUgeKENnaMeGayeFVItaragEdlmanBIrCYrQrKW TItaMgrbs;bnÞúkÉktþa
x nigtMélEdlKNnasMrab;kMlaMgRbtikmμ b¤kMlaMgkat; b¤m:Um:g;Bt; EdlRtUvnwgTItaMgenaH.

eRkayeBleKdak;bnÞúkenaHenAelITItaMgepSg²CaeRcIntambeNþayGgát; taragRtUv)anbMeBj
edaytMélelx ehIyeyIgeRbItMélkñúgtaragTaMgenaHedIm,Isg;düaRkamExST§iBl.
smIkarExST§iBl (influence-line equation)
eKk¾Gacsg;ExST§iBledaydak;bnÞúkÉktþaenATItaMg x elIGgát; rYcKNnatMél R / V b¤ M Ca
GnuKmn_ én x . sMrab;krNIenH eKGacsresr)ansmIkarénkMNat;Ggát;CaeRcInEdlbegáItCa
ExST§iBl rYceKGacsg;ExST§iBledayeRbIsmIkarTaMgenH.

]TahrN_ 6-1³ sg;ExST§iBlsMrab;kMlaMgRbtikmμbBaÄrrbs;FñwmenARtg; A ¬rUbTI 6-1a¦.
dMeNaHRsay³
CMnYstMélelxkñúgtarag³ bnÞúkÉktþaRtUv)andak;enAelIFñwmRtg;cMnuc x EdleRCIserIsnImYy² ehIytMél
Ay RtUv)anKNnaedayeFVIplbUkRtg;cMnuc B . ]TahrN_/ enAeBl x = 2.5m nig x = 5m emIlrUbTI

*
mUlehtuénkareRCIserIsenHRtUv)anbgðajenAkñúgkfaxNÐ 6.2.
Influence Lines for Statically Determinate Structures T.Chhay -169


6-1b nig 6-1c erogKña. lT§plsMrab; Ay RtUv)andak;bBa©ÚleTAkñúgtarag ¬rUbTI 6-1d¦. düaRkam
sMrab;tMélTaMgenH CaExST§iBlsMrab;RbtikmμenARtg;cMnuc A ¬rUbTI 6-1e¦.
smIkarExST§iBl³ enAeBleKdak;bnÞúkÉktþacMgayERbRbYl x BIcMnuc A ¬rUbTI 6-1f¦ eKGackMNt;
kMlaMgRbtikmμ Ay CaGnuKmn_én x BI
+ ∑MB = 0 − A y (10) + (10 − x )(1) = 0
1
Ay = 1 − x
10
smIkarbnÞat;enHRtUv)ansg;CadüaRkamenAkñúgrUbTI 6-1e.

]TahrN_ 6-2³ sg;ExST§iBlsMrab;kMlaMgRbtikmμbBaÄrenARtg; B rbs;FñwmenAkñúgrUbTI 6-2a.

dMeNaHRsay³
CMnYstMélelxkñúgtarag³ edayeRbIsþaTic epÞógpÞat;fatMélsMrab;Rbtikmμ B y EdlmanenAkñúgtarag ¬rUb


TI 6-2b¦ RtUv)anKNnay:agRtwmRtUvsMrab;TItaMg x énbnÞúkGefr. düaRkamrbs;tMélkñúgtaragCaExS
T§iBlsMrab;RbtikmμRtg; B ¬rUbTI 6-2c¦.
smIkarExST§iBl³ edayGnuvtþsmIkarm:Um:g;Rtg;cMnuc A ¬enAkñúgrUbTI 6-2d¦
+ ∑MA = 0 B y (5) − 1( x ) = 0
1
By = x
5
smIkarbnÞat;enHRtUv)ansg;CadüaRkamenAkñúgrUbTI 6-2c.

]TahrN_ 6-3³ sg;ExST§iBlsMrab;kMlaMgkat;enARtg;cMnuc C rbs;FñwmenAkñúgrUbTI 6-3a.

dMeNaHRsay³
CMnYstMélelxkñúgtarag³ enARtg;TItaMg x EdleRCIserIsnImYy²rbs;bnÞúkÉktþa eKeRbIviFImuxkat;edIm,I
KNnatMélrbs; VC . cMNaMfa eKRtUvdak;bnÞúkenAxageqVgcMnuc C bnþic ¬ x = 2.5− ¦ nigxagsþaMcMnuc C
bnþic ¬ x = 2.5+ ¦ edaysarkMlaMgkat;manlkçN³dac;enARtg;cMnuc C ¬rUbTI 6-3b nig 6-3c¦.


düaRkaméntMélenAkñúgrUbTI 6-3d CaExST§iBlsMrab;kMlaMgkat;enARtg;cMnuc C ¬rUbTI 6-3e¦.
smIkarExST§iBl³ enATIenH eKmansmIkarBIrEdlRtUvkMNt; edaysareKmankMNat;Ggát;BIrsMrab;ExS
T§iBlEdlbNþalBIPaBdac;rbs;kMlaMgkat;enARtg;cMnuc C ¬rUbTI 6-3f¦. smIkarTaMgenHRtUv)anKUr
düaRkamenAkñúgrUbTI 6-3e.


dMeNaHRsay³
CMnYstMélelxkñúgtarag³ edayeRbIsþaTic nigviFImuxkat; epÞógpÞat;fatMélrbs;kMlaMgkat;Vc enARtg;
cMnuc C enAkñúgrUbTI 6-4b EdlRtUvKñanwgTItaMg x nImYy²rbs;bnÞúkÉktþaenAelIFñwm. düaRkaméntMél
enAkñúgrUbTI 6-4b CaExST§iBlenAkñúgrUbTI 6-4c.
smIkarExST§iBl³ BIrUbTI 6-4d epÞógpÞat;fa
1
Vc = − x 0 ≤ x < 4m
8
1
Vc = 1 − x 4m < x ≤ 12m
8
smIkarTaMgenHRtUv)anbgðajCadüaRkamenAkñúgrUbTI 6-4c.




dMeNaHRsay³
CMnYstMélelxkñúgtarag³ enARtg;TItaMgeRCIserIsnImYy²rbs;bnÞúkÉktþa eKKNnatMélrbs; M C eday
eRbIviFImuxkat;. Ca]TahrN_ emIlrUbTI 6-5b sMrab; x = 2.5m . düaRkaméntMélenAkñúgrUbTI 6-5c
bgðajCaExST§iBlsMrab;m:Um:g;enARtg;cMnuc C ¬rUbTI 6-5d¦.
smIkarExST§iBl³ eKGaceRbIGgát;BIrkMNat;sMrab;kMNt;rksmIkarExST§iBledayeRbI ∑ M C = 0
CamYynwgviFImuxkat;EdlbgðajenAkñúgrUbTI 6-5d.
⎛ 1 ⎞
+ ∑MC = 0 M C + 1(5 − x ) − ⎜1 − x ⎟5 = 0
⎝ 10 ⎠



1
MC = x 0 ≤ x < 5m
2
⎛ 1 ⎞
+ ∑MC = 0 MC − ⎜1 − x ⎟5 = 0
⎝ 10 ⎠
1
MC =5− x 5m < x ≤ 10m
2


dMeNaHRsay³
CMnYstMélelxkñúgtarag³ edayeRbIsþaTic nigviFImuxkat; epÞógpÞat;fatMélrbs;m:Um:g; M c enARtg;cMnuc C
enAkñúgrUbTI 6-6b EdlRtUvKñanwgTItaMg x nImYy²rbs;bnÞúkÉktþaenAelIFñwm. düaRkaméntMélenAkñúgrUb
TI 6-6b CaExST§iBlenAkñúgrUbTI 6-6c.
smIkarExST§iBl³ BIrUbTI 6-6d epÞógpÞat;fa
1
MC = x 0 ≤ x < 4m
2
1
MC = 4 − x 4m < x ≤ 12m
2
smIkarTaMgenHRtUv)ansg;CadüaRkamdUcbgðajenAkñúgrUbTI 6-6c.



^>@> ExST§iBlsMrab;Fñwm (Influence line for beams)
edaysarFñwm ¬b¤r:t¦CaGgát;rgbnÞúkcMbgénRbB½n§kMral b¤kMrals<an eKcaM)ac;RtUvmanlT§PaB
sg;ExST§iBlsMrab;kMlaMgRbtikmμ kMlaMgkat; b¤m:Um:g;enARtg;cMnucCak;lak;enAkñúgFñwm.
bnÞúk³ enAeBleKsg;ExST§iBlsMrab;GnuKmn_ ¬kMlaMgRbtikmμ kMlaMgkat; b¤m:Um:g;¦ enaHeKnwgmanlT§-
PaBkMNt;TItaMgbnÞúkGefrenAelIFñwmEdlGacbegáItGnuKmn_mantMélGtibrma. eKmanbnÞúkBIrRbePT
EdlRtUvBicarNaenATIenH.
bnÞúkcMcMnuc³ edaysartMélCaelxénGnuKmn_sMrab;ExST§iBlRtUv)ankMNt;edayeRbIbnÞúkÉktþaKμanxñat
enaHsMrab;bnÞúkcMcMnuc F EdlmanGMeBIenAelIFñwmRtg;TItaMg x eKGacrktMélGnuKmn_edayKuNGredaen
rbs;ExST§iBlenARtg;TItaMg x edayTMhMrbs; F . Ca]TahrN_ eKmanExST§iBlsMrab;kMlaMgRbtikmμ
enARtg; A sMrab;Fñwm AB ¬rUbTI 6-7¦. RbsinebIbnÞúkÉktþasßitenARtg; x = L / 2 enaHkMlaMgRbtikmμ
enARtg; A KW Ay = 1 / 2 dUcbgðajenAkñúgExST§iBl. dUcenH RbsinebIkMlaMg F manGMeBIenARtg;TItaMgdUc
Kña kMlaMgRbtikmμrbs;vaKW Ay = F / 2 . Cak;Esþg eKGacTTYl)ankMlaMgRbtikmμEdlmantMéldUcKñaBI
karKNnaedaysþaTic. eyIgeXIjy:agc,as;fa T§iBlGtibrmaEdlbgáeday F ekItmanenAeBleK
dak;vaRtg;TItaMgénkMBUlrbs;ExST§iBl EdlenAkñúgkrNIenH x = 0 ehIyeyIgnwgTTYl)ankMlaMg
Rbtikmμ Ay = (1)(F ) .

bnÞúkBRgayesμI³ eKmanEpñkrbs;FñwmEdlrgbnÞúkBRgayesμI wo ¬rUbTI 6-8¦. dUckarbgðaj kMNat; dx
nImYy²rbs;bnÞúkenHbegáItkMlaMgcMcMnuc dF = wo dx enAelIFñwm. RbsinebI dF sßitenARtg; x Edl
GredaenénExST§iBlrbs;FñwmsMrab;GnuKmn_ ¬kMlaMgRbtikmμ kMlaMgkat; m:Um:g;¦xøH KW y enaHtMélén
GnuKmn_KW (dF )( y ) = (wo dx )y . T§iBlénkMlaMgcMcMnuc dF TaMgGs;RtUv)ankMNt;edaykareFVIGaMget
RkalelIRbEvgTaMgmUlrbs;Fñwm Edl ∫ wo ydx = wo ∫ ydx . ehIyedaysar ∫ ydx smmUleTAnwgRkLa
épÞExST§Bl enaHtMélrbs;GnuKmn_EdlbNþalBIbnÞúkBRgayesμInwgplKuNrvagRkLaépÞExST§iBlCa
mYynwgGaMgtg;sIuetrbs;bnÞúkBRgayesμI. Ca]TahrN_ enAkñúgkrNIFñwmrgbnÞúkBRgayesμIdUcbgðajenA


kñúgrUbTI 6-9 eKGackMNt;kMlaMgRbtikmμ Ay BIExST§iBlCa Ay = (area)(wo ) = [(1)(L) / 2]wo =
wo L / 2 . Cak;Esþg

eKGackMNt;tMélenHBIsþaTic.

]TahrN_ 6-7³ kMNt;kMlaMgkat;viC¢manGtibrmaEdlGacekItmanenARtg;cMnuc C enAkñúgFñwmdUcbgðaj
enAkñúgrUbTI 6-10a EdlbNþalBIbnÞúkcl½tcMcMnuc 4kN nigbnÞúkcl½tBRgay 2kN / m .

dMeNaHRsay³
sg;düaRkamExST§iBlsMrab;kMlaMgkat;sMrab;kMlaMgkat;Rtg;cMnuc C enAkñúg]TahrN_ 6-3 ehIyRtUv)an
bgðajenAkñúgrUbTI 6-10b.
bnÞúkcMcMnuc³ kMlaMgkat;viC¢manGtibrmaenARtg; C nwgekItmanenAeBlkMlaMg 4kN manGMeBIenARtg; x =
2.5+ m edaysarvakMBUlviC¢manrbs;ExST§iBl. GredaensMrab;kMBUlenHKW + 0.75 Edl

Vc = 0.75(4kN ) = 3kN



bnÞúkBRgay³ bnÞúkBRgaycl½tbegáItT§iBlviC¢manGtibrmasMrab;VC enAeBlEdlbnÞúkeFVIGMeBIenAelI
FñwmcenøaH x = 2.5+ m nig x = 10m edaysarenAkñúgtMbn;enH ExST§iBlmanRkLaépÞviC¢man. TMhMrbs;
VC EdlbNþalBIbnÞúkenHKW
⎡1 ⎤
VC = ⎢ (10m − 2.5m )(0.75)⎥ 2kN / m = 5.625kN
⎣2 ⎦
kMlaMgkat;GtibrmasrubenARtg;cMnuc C
(VC )max = 3kN + 5.625kN = 8.625kN
cMNaMfa enAeBlEdlTItaMgbnÞúkEdlbegáItedayeRbIExST§iBl ¬rUbTI 6-10c¦ tMél (VC )max enHk¾Gac
kMNt;edayeRbIsþaTic nigviFImuxkat;.

]TahrN_ 6-8³ eKeRbIeRKOgbgÁúMeRKagdUcbgðajenAkñúgrUbTI 6-11a edIm,IRT]bkrN_sÞÚc. bnÞúkEdl
]brkrN_sÞÚceyagmanbnÞúk 3kN ehIyFñwm CB manm:as 24kg / m . snμt; A CaTMrsnøak; ehIy B CaTMr
kl;. kMNt;kMlaMgRbtikmμTMrbBaÄrenARtg; A / B nigm:Um:g;GtibrmaenAkñúgFñwmRtg;cMnuc D .

dMeNaHRsay³
kMlaMgGtibrmaenARtg;cMnuc A ³ dMbUgeyIgKUrdüaRkamExST§iBlsMrab; Ay ¬rUbTI 6-11b¦. Cak;Esþg
enAeBlbnÞúkÉktþasßitenARtg; A kMlaMgRbtikmμenARtg; A esμInwg1.0 dUcbgðaj. GredaenenARtg; C esμI
nwg1.33 . enATIenHtMélGtibrmasMrab; Ay ekItmanenAeBlbnÞúksßitenARtg;cMnuc C . edaysarbnÞúk
Gefr ¬TMgn;Fñwm¦ manelIRbEvgFñwmTaMgmUl eyIg)an


(Ay )max = 3000(1.33) + 24(9.81)⎡ 1 (4)(1.33)⎤
⎢2 ⎥
⎣ ⎦

= 4.63kN
kMlaMgGtibrmaenARtg;cMnuc B ³ ExST§iBl ¬b¤Fñwm¦ manragdUcbgðajenAkñúgrUbTI 6-11c. eKkMNt;
tMélenARtg; C nig B edaysþaTic. enATIenHbnÞúkRtUvsßitenARtg;cMnuc B . dUcenH
(B y )max = 3000(1) + 24(9.81)⎡ 1 (3)(1)⎤ + 24(9.81)⎡ 1 (1)(− 0.333)⎤
⎢2 ⎥ ⎢2 ⎥
⎣ ⎦ ⎣ ⎦

= 3.31kN
m:Um:g;GtibrmaenARtg;cMnuc D ³ ExST§iBlmanrUbragdUcbgðajenAkñúgrUbTI 6-11d. eKkMNt;tMélenA
Rtg; C nig D BIsmIkarsþaTic. enATIenH
(M D )max = 3000(0.75) + 24(9.81)⎡ 1 (1)(− 0.5)⎤ + 24(9.81)⎡ 1 (3)(0.75)⎤
⎢ ⎥ ⎢ ⎥
⎣2 ⎦ ⎣2 ⎦

= 2.46kN.m

^>#> eKalkarN_muyeyeRbsøÚ (Qualitative influence lines)
enAqñaM 1886/ Heinrich Muller Breslau )anbegáItbec©keTssMrab;sg;ExST§iBly:agelOn
bMput. eKakarN_muyeyeRbsøÚeBalfaExST§iBlsMrab;GnuKmn_kMlaMg ¬kMlaMgRbtikmμ kMlaMgkat; b¤
m:Um:gBt;¦ manragdUcnwgrUbragdabrbs;FñwmenAeBlFñwmrgGMeBIedayGnuKmn_kMlaMg. edIm,IKUrrUbragdab
eGay)anl¥ eKRtUvyklT§PaBrbs;FñwmTb;Tl;GnuKmn_kMlaMg dUcenHFñwmGacdabenAeBlGnuKmn_kMlaMg
TaMgenHmanGMeBI.

Ca]TahrN_ eKmanFñwmenAkñúgrUbTI 6-12a. RbsinebIeKRtUvkarkMNt;ragrbs;ExS T§iBlsMrab;
kMlaMgRbtikmμRtg;cMnuc A enaHeKRtUvCMnYssnøak;edaypøÚvr:UlU (roller guide) dUcbgðajenAkñúgrUbTI 6-
12b. eK RtUvkarpøÚvr:UlU (roller guide) edaysarFñwmRtUvTb;kMlaMgedkenARtg;cMnuc A minEmnkMlaMg


bBaÄr. enAeBleKGnuvtþ kMlaMgviC¢man ¬TisedAeLIgelI¦ Ay enARtg;cMnuc A FñwmxUcRTg;RTaydUcbnÞat;
dac; EdlbgðajrUbrag TUeTArbs;ExST§iBlsMrab; Ay ¬rUbTI 6-12c¦. tMélCaelxsMrab;krNICak;
*

EsþgRtUv)anKNnaenAkñúg]TahrN_ 6-1.

RbsinebIeKRtUvkarkMNt;rUbragrbs;ExST§iBlsMrab;kMlaMgkat;enARtg; cMnuc C ¬rUbTI 6-13a¦
eKGaceFVInimitþsBaØapøÚvr:UlU (roller guide) Rtg;cMnuc C dUcbgðajenAkñúgrUbTI 6-13b. ]bkrN_enHnwg
Tb;Tb;m:Um:g; nig kMlaMgtamG½kS EtminTb;Tl;nwgkMlaMgkat;eT . edayGnuvtþkMlaMgkat;viC¢man VC eTA
†

elIFñwmRtg;cMnuc C nigedayGnuBaØateGayFñwmdabdUcTItaMgbnÞat;dac; enaHExST§iBlsMrab;kMlaMgkat;
Rtg;cMnuc C manragdUcbgðajenAkñúgrUbTI 6-13c. cugeRkay RbsinebIeKRtUvkarkMNt;ragExST§iBl
sMrab;m:Um:g;Rtg;cMnuc C ¬rUbTI 6-14a¦ eKRtUvdak;snøak;enARtg;cMnuc C edaysartMNenHTb;Tl;kMlaMg
tamG½kS nigkMlaMgkat; EtminTb;Tl;nwgm:Um:g;eT ¬rUbTI 6-14b¦. edayGnuvtþm:Um:g;viC¢man M C eTAelI
FñwmRtg;cMnuc C Fñwmnwg dabdUcTItaMgbnÞat;dac; EdlCaragExST§iBlsMrab;m:Um:g;Bt;Rtg;cMnuc C ¬rUbTI 6-
13d¦.

*
RKb;karBnül;TaMgGs;rUbragdabRtUv)anKUredaymaRtdæanFMCagedIm,IbgðajBIeKalKMnit.
†
enATIenHr:UlUCanimitþsBaØaTMrEdlTb;Tl;bnÞúkTaMgkarTaj nigTaMgkarsgát; ¬emIltarag 2-1 TMr (2)¦.


eKGacbkRsayeKalkarN_muyeyeRbsøÚedayeRbIeKalkarN_ virtual work. rMlwkfakmμnþ
(work) CaplKuNrvagcMgaybMlas;TICamYykMlaMgEdlmanGMeBIRsbtamTisrbs;bMlas;TI b¤Capl

KuNrvagmMurgVilnigm:Um:g;EdlmanGMeBIRsbtamTisrbs;mMurgVil. RbsinebIGgÁFaturwg ¬Fñwm¦ sßitkñúgsßan
PaBlMnwg plbUkénkMlaMgTaMgGs; nigplbUkm:Um:g;TaMgGs;EdlmanGMeBIelIFñwmRtUvEtesμIsUnü. dUcenH
RbsinebIGgÁFatupþl;nUvbMlas;TInimitþ b¤ bMlas;TI virtual/ kmμnþEdlekIteLIgedaykMlaMg nigm:Um:g;TaMg
enHk¾RtUvEtesμInwgsUnüEdr. Ca]TahrN_ eKmanFñwmTMrsamBaØ ¬rUbTI 6-15a¦ EdlrgbnÞúkÉktþaEdl
dak;enARtg;cMnucNamYytambeNþayRbEvg. RbsinebIFñwmmanbMlas;TInimitþ (virtual displacement)
δy enARtg;TMr A ¬rUb 6-15b¦ manEtkMlaMgRbtikmμ Ay nigbnÞúkÉktþabegáItkmμnþnimitþ. CaBiess Ay
begáItkmμnþviC¢man Ayδy nigbnÞúkÉktþabegáItkmμnþGviC¢man − 1δy' . TMrenARtg; B mincl½t ehIykMlaMg
enARtg; B minbegáItkmμnþ. edaysarFñwmsßitenAkñúgsßanPaBlMnwg dUcenHmincl½tCak;lak; plbUk-
kMlaMgnimitþesμIsUnü Edl
Ay δy − 1δy ' = 0

RbsinebIeKeGay δy = 1 enaH
Ay = δy '

niyaymüa:geTot tMélrbs; Ay CakUGredaenrbs;ExST§iBlenARtg;TItaMgrbs;bnÞúkÉktþa. edaysar
tMélenHsmmUleTAnwgbMlas;TI δy' enARtg;TItaMgrbs;bnÞúkÉktþa vabgðajfarUbragrbs;ExST§iBlsMrab;
kMlaMgRbtikmμenARtg; A RtUv)anbegáIt. TaMgenHbgðajBIeKalkarN_muyeyeRbsøÚsMrab;kMlaMgRbtikmμ.

enAkñúgrebobdUcKña RbsinebIFñwmRtUv)ankat;enARtg; C ehIyFñwmrgbMlas;TI δ y enARtg;cMnucenH
¬rUbTI 6-15c¦ enaHkMlaMgkat;enARtg;cMnuc C ehIybnÞúkÉktþabegáItkmμnþ. dUcenH smIkarkmμnþnimitþKW
VC δy − 1δy ' = 0



mþgeTot RbsinebI δy = 1 enaH
VC = δy '
ehIyeK)anbegáItrUbragrbs;ExST§iBlsMrab;kMlaMgkat;enARtg;cMnuc C .
cugeRkay snñt;snøak;RtUv)anbegáItenAkñúgFñwmRtg;cMnuc C ¬rUbTI 6-15d¦. RbsinebImMurgVil δφ
RtUv)anbegáItenARtg;snøak; kmμnþnimitþRtUv)anbegáItedaym:Um:g;xagkñúg nigbnÞúkÉktþa. dUcenH
M C δφ − 1δy ' = 0
edayyk δφ = 1 eyIgeXIjfa
M C = δy '
EdlbgðajfaFñwmmanPaBdabdUcragCaExST§iBlsMrab;m:Um:g;xagkñúgenARtg;cMnuc C ¬emIlrUbTI 6-14¦.
eyIgeXIjy:agc,as;fa eKalkarN_muyeyeRbsøÚpþl;nUvviFIy:agelOnsMrab;begáItrUbragrbs;ExS
T§iBl. enAeBlEdleyIgsÁal;ragrbs;vaehIy eyIgGackMNt;GredaenenARtg;kMBUledayeRbIviFImUl-
dæanEdlerobrab;enAkñúgkfaxNÐ 6-1. karsÁal;rUbragTUeTArbs;ExST§iBl eFVIeGayeyIgGacmanlT§-
PaBkMNt;TItaMgbnÞúkGefrenAelIFñwm nigbnÞab;mkeyIgGackMNt;tMélGtibrmarbs;GnuKmn_kMlaMgeday
eRbIsmIkarsþaTic. ]TahrN_ 6-12 bgðajBIbec©keTsenH.

]TahrN_ 6-9³ sMrab;FñwmnImYy²enAkñúgrUbTI 6-16a rhUtdl; 6-16c KUrExST§iBlsMrab;kMlaMg
RbtikmμbBaÄrenARtg; A .



dMeNaHRsay³
eKCMnYsTMr A edaypøÚvr:UlU (roller guide) nigGnuvtþkMlaMg Ay sMrab;rUb 6-16a.
mþgeTot CMnUspøÚvr:UlU (roller guide) Rtg;cMnuc A ehIyGnuvtþkMlaMg Ay eTAelIva ¬rUbTI 6-16b¦.
eKRtUveRbIpøÚvr:UlUBIr (two-roller guide)Rtg;cMnuc A sMrab;krNIrUbTI 6-16c
edaysarTMrRbePTenHbBa¢ÚnTaMgm:Um:g; M A nigbnÞúktamG½kS Ax EtminbBa¢Ún Ay .

]TahrN_ 6-10³ sMrab;FñwmnImYy²enAkñúgrUbTI 6-17a rhUtdl; 6-17c KUrExST§iBlsMrab;kMlaMg
kat;enARtg;cMnuc B .

dMeNaHRsay³
eKdak;pøÚvr:UlU (roller guide) enARtg;cMnuc B nigGnuvtþkMlaMgkat;viC¢man VB ¬rUb 6-17a¦. cMNaMfa
kMNat;Ggát;xagsþaMrbs;Ggát;mindabeT edaysarTMrkl;enARtg; A Tb;FñwmmineGayeFVIclnatamTis
bBaÄr ¬emIlTMr (2) enAkñúgtarag 2-1¦.
edaydak; roller guide Rtg;cMnuc B ehIyGnuvtþkMlaMgkat;viC¢man enARtg;cMnuc B eKTTYl)anragdab
rbs;FñwmEdlRtUvKñanwgExST§iBl ¬rUbTI 6-17b¦.
eKRtUvdak;pøÚvr:UlURtg;cMnuc B ehIyGnuvtþkMlaMgkat;viC¢man enaHeKnwg)anrUbragxUcRTg;RTayrbs;Fñwm
EdlRtUvKñanwgExST§iBl ¬rUbTI 6-17c¦. cMNaMfakMNat;Ggát;xageqVgrbs;Fñwmmindab edaysarTMr
bgáb;.



]TahrN_ 6-11³ sMrab;FñwmnImYy²enAkñúgrUbTI 6-18a rhUtdl; 6-18c KUrExST§iBlsMrab;m:Um:g;Bt;
enARtg;cMnuc B .

dMeNaHRsay³
eKdak;snøak;enARtg;cMnuc B ehIyGnuvtþm:Um:g;viC¢man M B eTAelIFñwm ¬rUb 6-18a¦. ragdabrbs;Fñwm
nigExST§iBlRtUv)anbgðajenAkñúgrUbTI 6-18a.
edaydak;snøak;enARtg;cMnuc B ehIyGnuvtþm:Um:g;viC¢man M B eTAelIFñwm eKnwgTTYl)anragdabrbs;Fñwm
EdlRtUvKñanwgExST§iBl ¬rUbTI 6-18b¦.
CamYynwgsnøak; nigm:Um:g;viC¢manenARtg; B rUbragxUcRTg;RTayrbs;FñwmEdlRtUvKñanwgExST§iBlRtUv)an
bgðajenAkñúgrUbTI 6-18c. cMNaMfakMNat;Ggát;xageqVgrbs;FñwmRtUv)anTb;mineGayviledaysarTMr
bgáb;.

]TahrN_ 6-12³ kMNt;m:Um:g;viC¢manGtibrmaEdlGacekItmanenARtg;cMnuc D enAkñúgFñwmEdlbgðaj
enAkñúgrUbTI 6-19a EdlbNþalBIbnÞúkcl½tcMnuc16kN bnÞúkcl½tBRgay 3kN / m nigTMgn;Fñwm
2kN / m .

dMeNaHRsay³ eKdak;snøak;enARtg;cMnuc D ehIyGnuvtþm:Um:g;viC¢man M D eTAelIFñwm. rUbragdab nig
ExST§iBlEdlRtUvKñaRtUv)anbgðajenAkñúgrUbTI 6-19b. eyIgdwgehIyfa bnÞúkcl½tcMcMnuc16kN begáIt
m:Um:g;viC¢manGtibrmaenARtg;cMnuc D enAeBleKdak;vaenARtg;cMnuc D eBalKWenARtg;kMBUlrbs;ExST§i-
Bl. ehIybnÞúkcl½tBRgay 3kN / m RtUv)andak;BRgayBIcMnuc C eTAcMnuc E edIm,IRKbelItMbn;Edl
RkLaépÞrbs;ExST§iBlmantMélviCman. cugeRkay TMgn;rbs;Fñwm 2kN / m manGMeBIelIRbEvgFñwmTaMg
¢


mUl. bnÞúkEdlmanGMeBIelIFñwmRtUv)anbgðajenAkñúgrUbTI 6-19c. edaysÁal;TItaMgrbs;bnÞúk eyIgGac
kMNt;m:Um:g;GtibrmaenARtg; D edayeRbIsmIkarsþaTic. enAkñúgrUbTI 6-19d kMlaMgRbtikmμenAelI BE
RtUv)ankMNt;. edaykat;FñwmenARtUv)ancMnuc D nigedayeRbIkMNat;Ggát; DE ¬rUbTI 6-19e¦ eyIg)an



+ ∑MD = 0 − M D − 20(2 ) + 19(4 ) = 0

M D = 36kN .m
eyIgk¾GacedaHRsaycMeNaTenHedayeRbItMélCaelxsMrab;ExST§iBldUcenAkñúgkfaxNÐ 6-1.
edaykarGegátrUbTI 6-19b Cak;Esþg eKRtUvkarKNnaEttMélkMBUl h Rtg; D b:ueNÑaH. dUcenHvaTamTar
eGaydak;bnÞúkÉktþaenAelIFñwmRtg;cMnuc D kñúgrUbTI 6-19a nigbnÞab;mkedaHRsaym:Um:g;Bt;enAkñúgFñwm
Rtg;cMnuc D . bgðajfatMélEdlTTYl)anKW h = 1.333 . edayRtIekaNdUc h' /(4 − 2) = 1.33 /(6 − 4) b¤
h' = 1.333 . dUcenH CamYynwgbnÞúkenAelIFñwmdUcenAkñúgrUbTI 6-19c edayeRbIRkLaépÞ nigtMélkMBUlén

ExST§iBl ¬rUbTI 6-19b¦ eyIg)an
⎡1 ⎤ ⎡1 ⎤
M D = 5⎢ (10 − 4 )(1.333)⎥ + 16(1.333) − 2 ⎢ (4)(1.333)⎥
⎣2 ⎦ ⎣2 ⎦

= 36kN.m

^>$> ExST§iBlsMrab;r:tRTkMral (Influence lines for floor Girders)
eBlxøH RbB½n§kMralRtUv)ansg;dUcbgðajenAkñúgrUbTI 6-20a EdleyIgeXIjfabnÞúkkMral
(floor load) RtUv)anepÞrBIkMralxNÐeTAFñwmkMral nigbnÞab;mkepÞreTAr:txag ehIycugeRkayepÞreTA

ssrTMr. KMrUm:UEDlénRbB½n§enHRtUv)anbgðajCaKMeXIjkñúgbøg;kñúgrUbTI 6-20b. enATIenH eKsnμt;fa
kMralxNÐCakMralmYyTis ehIyRtUv)anbMEbkCaElVgTMrsamBaØEdlQrelIFñwmkMral. ehIy r:tRtUv)an
RTedayTMrsamBaØenAelIssr. edaysarr:tCaGgát;rgbnÞúkcMbgenAkñúgRbB½n§enH eBlxøHeKcaM)ac;sg;
ExST§iBlsMrab;kMlaMgkat; nigExST§iBlsMrab;m:Um:g;rbs;va sMrab;GKar]sSahkmμEdlrgbnÞúkcMcMnucF¶n;.
cMNaMfa bnÞúkÉktþaenAelIkMralxNÐRtUv)anepÞreTAr:tEtenARtg;cMnucRbsBVrvagFñwmkMral nigr:t eBalKW
cMMnuc A / B / C nig D . cMnucTaMgenHeKehAfa panel point ehIytMbn;cenøaHcMnucTaMgenHeKeGayeQμaH
fa panel dUcCa BC enAkñúgrUbTI 6-20b.
eKGackMNt;ExST§iBlsMrab;cMnucCak;lak;enAelIr:tedayeRbIdMeNIrkarsþaTicdUcKñaenAkñúgkfa-
xNÐ 6-1 eBalKW dak;bnÞúkÉktþaenAelIcMnuc x NamYyenAelIkMralxNÐ nigkMNt;GnuKmn_kMlaMg
¬kMlaMgkat; b¤m:Um:g;¦ enARtg;cMnucCak;lak; P enAkñúgr:t ¬rUbTI 6-20b¦. sg;düaRkaméntMélTaMgenH
GaRs½ynwg x enaHeyIgnwgTTYl)anExST§iBlsMrab;GnuKmn_kMlaMgenARtg; P . CaBiess tMélsMrab;
m:Um:g;enAkñúg girder panel EtgGaRs½ynwgkareRCIserIsTItaMgrbs; P sMrab;ExST§iBl edaysarTMhM
M P GaRs½ynwgTItaMgrbs;cMnucBIcugrbs;r:t. Ca]TahrN_ RbsinebIbnÞúkÉktþamanGMeBIenAelIkMral

xNÐdUcbgðajenAkñúgrUbTI 6-20c dMbUgeKGacrkkMlaMgRbtikmμ FB nig FC enAelIkMral bnÞab;mk


KNnakMlaMgRbtikmμ F1 nig F2 enAelIr:t. bnÞab;mkkMNt;m:Um:g;Bt;enARtg; P edayviFImuxkat; ¬rUbTI
6-20d¦. CalT§pleyIgnwgTTYl)an M P = F1d − FB (d − s ) . edayeRbIkarviPaKdUcKña eyIgnwg
TTYl)an VP mantMélefrelI panel BC TaMgmUl Edl VP = F1 − FB dUcenHvaminGaRs½ynwgTItaMgCak;
lak; d rbs; P enAkñúg panel eT. sMrab;mUlehtuenH ExST§iBlsMrab;kMlaMgkat;rbs;kMralr:tRtUv)an
kMNt;sMrab; panel minEmnRtg;cMnucNamYytambeNþayr:teT. kMlaMgkat;enAkñúgr:tRtUv)aneKeGay
eQμaHfa panel shear. eKRtUvcMNaMfa edaysarr:trgT§iBledaysarEtbnÞúkEdlepÞredayFñwmkMral
CaeKTUeTAeKeRcIndak;bnÞúkÉktþaenARtg;TItaMgFñwmkMralnImYy²edIm,IbegáItTinñn½ycaM)ac;edIm,IKUrExST§i-
Bl.
]TarhrN_CatMélelxxageRkamnwgbgðajy:agc,as;BIkarviPaKkMlaMg.

]TahrN_ 6-13³ sg;ExST§iBlsMrab;kMlaMgkat;enAkñúg panel CD sMrab;r:tRTkMralenAkñúgrUbTI 6-
21a.
dMeNaHRsay³
CMnYstMélelxkñúgtarag³ eKdak;bnÞúkÉktþaenARtg;TItaMgFñwmkMralnImYy² ehIyKNnakMlaMgkat;enAkñúg
panel CD . taragénlT§plRtUv)anbgðajenAkñúgrUbTI 6-21b. karKNnalMGitenAeBl x = 0 nig x

= 6m RtUv)aneGayenAkñúgrUbTI 6-21c nig 6-21d erogKña. cMNaMfa dMbUgeKRtUvkMNt;kMlaMgRbtikmμ



rbs;FñwmkMralenAelIr:tkñúgkrNInImYy² bnÞab;mkkMNt;kMlaMgRbtikmμrbs;r:tenARtg;cMnuc F ¬eKmin
RtUvkar G y eT¦ ehIycugeRkay eKBicarNakMNat;Ggát;rbs;r:t ehIyKNnakMlaMgkat; VCD enAkñúg
panel CD . sMrab;kic©karbEnßm cUrepÞógpÞat;tMél VCD enAeBl x = 3m / 9m nig 12m .

ExST§iBl³ enAeBlEdlelxCMnYskñúgtaragRtUv)ansg;CadüaRkamedayP¢ab;BIcMnucmYyeTAcMnucmYy
edaybnÞat;Rtg; lT§plénExST§iBlsMrab; VCD RtUv)anbgðajenAkñúgrUbTI 6-21e.

]TahrN_ 6-14³ sg;ExST§iBlsMrab;m:Um:g;enARtg;cMnuc F sMrab;r:tRTkMralenAkñúgrUbTI 6-22a.
dMeNaHRsay³
CMnYstMélelxkñúgtarag³ eKdak;bnÞúkÉktþaenARtg; x = 0 nigenARtg; panel point nImYy². tMélRtUvKña
sMrab; M F RtUv)anKNna ehIyRtUv)anbgðajenAkñúgtarag ¬rUbTI 6-22b¦. karKNnalMGitsMrab; x =
2m RtUv)anbgðajenAkñúgrUbTI 6-22c. dUcenAkñúg]TahrN_elIkmun dMbUgeKcaM)ac;kMNt;kMlaMgRbtikmμ

rbs;FñwmkMralEdlenAelIr:t bnÞab;mkkMNt;kMlaMgRbtikmμ G y rbs;r:t ¬eKmincaM)ac;kMNt; H y eT¦ehIy



cugbBa©b; eKBicarNakMNat;Ggát; GF rbs;r:t nigKNnam:Um:g;kñúg M F . sMrab;kic©karbEnßm cUrkMNt;
tMélepSgeTotrbs; M F EdlmanenAkñúgtaragkñúgrUbTI 6-22b.
ExST§iBl³ düaRkaméntMélkñúgtaragCaExST§iBlsMrab; M F RtUv)anbgðajenAkñúgrUbTI 6-22d.

^>%> ExSTi§BlsMrab; trusses (Influence lines for trusses)
eBlxøHeKeRbI truss CaeRKOgbgÁúMRTbnÞúkcMbgsMrab;s<an. dUcenH sMrab;karsikSaKNna eKRtUv
manlT§PaBsg;ExST§iBlsMrab;Ggát;rbs;vanImYy². dUcbgðajenAkñúgrUbTI 6-23 bnÞúkenAelIkMral
s<anRtUv)anepÞreTA stringer EdlvabBa¢ÚnbnþeTAFñwmkMral nigbnÞab;mkbnÞúkRtUv)anbBa¢ÚneTAtMNénGgát;
xageRkam (bottom cord) rbs; truss. edayGgát; truss rgT§iBlEtBIbnÞúkRtg;tMN dUcenHeyIgGac
TTYltMélGredaenrbs;ExST§iBlsMrab;Ggát;edaykardak;bnÞúkÉktþaenARtg;tMNnImYy²tambeNþaykMr
al ehIyeRbIviFItMN b¤viFImuxkat;edIm,IKNnakMlaMgenAkñúgGgát;. eKGactMerobTinñn½yenAkñúgtarag Edl
manbnÞúkÉktþa nigkMlaMgkñúgrbs;Ggát;. edIm,IPaBgayRsYy RbsinebIGgát;rgkMlaMgTaj eKKitfava
mantMélviC¢man RbsinebIrgkMlaMgsgát; vamantMélGviC¢man. ExST§iBlsMrab;Ggát;RtUv)ansg;eday
bgðajTinñn½yenAkñúgCacMnuc ehIyKUrP¢ab;BIcMnucmYyeTAcMnucmYyedaybnÞat;Rtg;.


]TahrN_xageRkambgðajBIviFIkñúgkarsg;ExST§iBl.

]TahrN_ 6-15³ sg;ExST§iBlsMrab;kMlaMgkñúgGgát; GB sMrab; truss s<anEdlbgðajenAkñúgrUbTI
6-24a.

dMeNaHRsay³
CMnYstMélelxkñúgtarag³ enATIenH eKdak;bnÞúkÉktþaenARtg;tMNnImYy²enARtg;Ggát;xageRkambnþbnÞab;


rYcKNnakMlaMgenAkñúgGgát; GB edayeRbIviFImuxkat; ¬rUbTI 6-24b¦. Ca]TahrN_ edaydak;bnÞúk
ÉktþaenARtg; x = 6m ¬tMN B ¦ dMbUgKNnakMlaMgRbtikmμTMrenARtg; E ¬rUbTI 6-24a¦ bnÞab;mkkat;
truss enARtg;Ggát; HG / GB / BC ehIykMNt;kMlaMgenAkñúg GB BIkMNat; truss xagsþaM ¬rUbTI 6-24c¦.

tamrebobdUcKña kMNt;tMéldéTepSgeTotEdlmanenAkñúgtarag.
ExST§iBl³ edaybgðajtMélkñúgtaragCacMnucenAelIkUGredaen nigedayP¢ab;cMnucnImYy²edaybnÞat;Rtg;
eyIg)anExST§iBlsMrab;Ggát; GB ¬rUbTI 6-24d¦. edaysarExST§iBlbnøayelIElVgrbs; truss
TaMgRsug enaHeKcat;TukGgát; GB CaGgát;em (primary member). enHmann½yfa GB rgkMlaMgeday
minKitTItaMgeFVIGMeBIrbs;bnÞúkmkelIkMrals<aneT. cMnucénkMlaMgsUnü ¬ x = 8m ¦ RtUv)ankMNt;eday
RtIekaNdUcrvag x = 6m nig x = 12m Edl (0.354 + 0.707) /(12 − 6) = 0.354 / x' ¬ x' = 2m ¦ dUcenH
x = 6 + 2 = 8m .

]TahrN_ 6-16³ sg;ExST§iBlsMrab;kMlaMgkñúgGgát; CG sMrab; truss s<anEdlbgðajenAkñúgrUbTI
6-25a.

dMeNaHRsay³
CMnYstMélelxkñúgtarag³ taragénbnÞúkÉktþaenARtg;tMNénGgát;xageRkam nigkMlaMgxagkñúgGgát; CG
RtUv)anbgðajenAkñúgrUbTI 6-25b. eyIgGacTTYl)antMélTaMgenHy:agRsYlbMputedaypþac;tMN C ¬rUb
TI 6-25c¦. enATIenH eyIgeXIjfaGgát; CG CaGgát;EdlmankMlaMgsUnü elIkElgEtkñúgkrNIbnÞúk


ÉktþamanGMeBIenARtg;tMN C EdlenAkñúgkrNIenH FCG = 1(T ) .
ExST§iBl³ edaybgðajtMélkñúgtaragCacMnucenAelIkUGredaen nigedayP¢ab;cMnucnImYy²edaybnÞat;Rtg;
eyIg)anExST§iBlsMrab;Ggát; CG ¬rUbTI 6-25d¦. cMNaMfa enAeBlbnÞúkÉktþasßitenARtg; x = 9m
kMlaMgenAkñúg CG KW FCG = 0.5 . sßanPaBEbbenHTamTareGaydak;bnÞúkÉktþaenAelIkMrals<ancenøaH
tMN. karepÞrbnÞúkBIkMraleTA truss RtUv)anbgðajenAkñúgrUbTI 6-25e. BIrUbenH eyIgeXIjfa FCG =
0.5 edaykarviPaKlMnwgéntMN C ¬rUbTI 6-25f¦. edaysarExST§iBlsMrab; CG minlatsn§wgelI

ElVgTaMgmUlrbs; truss ¬rUbTI 6-25d¦ eKehAGgát; CG CaGgát;rg (secondary member).

]TahrN_ 6-17³ kMNt;kMlaMgFMbMputEdlGacekItmanenAkñúgGgát; BC rbs; truss s<anEdlbgðaj
enAkñúgrUbTI 6-26a EdlbNþalBIbnÞúkcl½t100kN nigbnÞúkBRgaycl½t 7.5kN / m . bnÞúkRtUv)an
GnuvtþenAGgát;xagelI.

dMeNaHRsay³
CMnYstMélelxkñúgtarag³ taragénTItaMgbnÞúkÉktþa x enARtg;tMNtambeNþayGgát;xagelICamYynwg
kMlaMgenAkñúgGgát; BC RtUv)anbgðajenAkñúgrUbTI 6-26b. eKGaceRbIviFImuxkat;sMrab;karKNna. Ca
]TahrN_ enAeBlbnÞúkÉktþaenARtg;cMnuc I Edl x = 8m dUcbgðajenAkñúgrUbTI 6-26a dMbUgeKRtUv
kMNt;kMlaMgRbtikmμ E y Edl E y = 0.25 . bnÞab;mk eKkat; truss Rtg;Ggát; BC / HB nig HI ehIy



eyIgBinitükMNat; truss xagsþaM ¬rUbTI 6-26c¦. eKTTYl FBC edayeFVIplbUkm:Um:g;Rtg;cMnuc H edIm,I
lubbM)at; FHI nig FHB . tamrebobdUcKña kMNt;tMéldéTeTotenAkñúgrUbTI 6-26b.
ExST§iBl³ düaRkaméntMélkñúgtaragCaExST§iBl ¬rUbTI 6-26d¦. tamkarGegát Ggát; BC CaGgát;
em. ehtuGVI?
kMlaMgcMcMnucGefr³ bnÞúkFMbMputenAkñúgGgát; BC ekItmanenAeBleKdak;kMlaMgcl½t100kN enARtg;
x = 16m . dUcenH

FBC = (1.33)(100) = 133kN
bnÞúkBRgayGefr³ eKRtUvdak;bnÞúkBRgayGefrenAelI truss TaMgmUledIm,IbegáIt)ankMlaMgTajFMbMput
enAkñúgGgát; BC . dUcenH
*

⎡1 ⎤
FBC = ⎢ (32)(1.33)⎥ 7.5 = 160kN
⎣2 ⎦
kMlaMgGtibrmasrub
(FBC )max = 1.33kN + 160kN = 293kN

^>^> T§iBlGtibrmaenARtg;cMnucmYyEdlbNþalBIes‘rIénbnÞúkcMcMnuc
(Maximum influence at a point due to a series of concentrated loads)
enAeBleKsg;ExST§iBlrbs;GnuKmn_kMlaMgsMrab;cMnucmYyenAkñúgeRKOgbgÁúMrUcehIy eKGackM-
Nt;T§iBlGtibrmaEdlbNþalBIkMlaMgcMcMnucGefredayKuNGredaenkMBUlrbs;ExST§iBlCamYynwgTM
hMrbs;kMlaMg. b:uEnþenAkñúgkrNIxøH eRKOgbgÁúMEtgEtrgnUves‘rIénbnÞúkcMcMnuc dUcCabnÞúkrbs;kg;rfynþdwk
TMnij b¤rfePøIg. edIm,IkMNt;T§iBlGtibrmaenAkñúgkrNIEbbenH eKGaceRbIviFIsakl,gEktMrUv (trial-
and-error procedure) b¤eRbIviFIEdlQrelIkarbMErbMrYlGnuKmn_kMlaMgEdlKitfabnÞúkcl½t. enATIenH

viFInImYy²RtUv)anBnül;sMrab;karGnuvtþeTAelIkMlaMgkat; nigm:Um:g;Bt;.
kMlaMgkat;³ eKmanFñwmTMrsamBaØCamYynwgExST§iBlsMrab;kMlaMgkat;enARtg;cMnuc C enAkñúgrUbTI 6-27a.
eKGackMNt;kMlaMgkat;viC¢manGtibrmaenARtg;cMnuc C EdlbNþalBIes‘rIénbnÞúkcMcMnucEdlcl½telIFñwmBI
sþaMeTAeqVg. bnÞúkeRKaHfñak;ekItmanenAeBleKdak;bnÞúkmYyenAxagsþaMcMnuc C bnþicEdlRtYtsIuKñanwgcMnuc
kMBUlrbs;ExST§iBl. eday trial and error eKGaceFVIkarGegátkrNIEdlGacekItmancMnYnbI ¬rUbTI 6-
27b¦. eyIg)an
*
kMlaMgTajFMbMputenAkñúgGgát; GB én]TahrN_ 6-15 ekItmanenAeBlbnÞúkBRgayeFVIGMeBIenAelI truss BI x = 0m eTA 8m
¬rUbTI 6-24d¦


krNITI1³ (VC )1 = 4.5(0.75) + 18(0.625) + 18(0.5) = 23.63kN
krNITI2³ (VC )2 = 4.5(− 0.125) + 18(0.75) + 18(0.625) = 24.19kN
krNITI3³ (VC )3 = 4.5(0) + 18(− 0.125) + 18(0.75) = 11.25kN
krNITI2 EdlmankMlaMg 4.5kN sßitenARtg;1.5+ m BITMrxageqVgeFVIeGaytMélVC FMbMput dUcenH
vaCakardak;bnÞúkEdleRKaHfñak;bMput. Cak;Esþg karGegátkrNITI3 KWmanlkçN³mincaM)ac; edaysar
kartMerobbnÞúkEbbenHeFVIeGaytMélrbs; (VC )3 tUcCag (VC )2 .

enAeBlbnÞúkcMcMnucCaeRcIneFVIGMeBIelIElVgFñwm dUckñúgkrNIbnÞúk E-72 énrUbTI 1-11 eKGaceRbI
karKNna trial-and-error xagelI)an. eKGackMNt;TItaMgeRKaHfñak;rbs;bnÞúkedayrkbMErbMrYlkMlaMg
kat; ΔV EdlekItmanenAeBlbnÞúkcl½tBIkrNITImYyeTAkrNITIBIr bnÞab;mkBIkrNITIBIreTAkrNITIbI
nigbnþbnÞab;. enAeBlEdlkarKNna ΔV viC¢man TItaMgfμInwgeFVIeGaymankMlaMgkat;enAkñúgFñwmRtg; C FM
CagTItaMgelIkmun. eKRtUveFVIkarGegátrhUtdl;TTYl)anbMErbMrYlkMlaMgkat;GviC¢man. enAeBleyIg
TTYl)antMélGviC¢manEbbenH mann½yfaTItaMgrbs;bnÞúkelIkmunpþl;nUvtMéleRKaHfñak;. eKGackMNt;
bMErbMrYlkMlaMgkat; ΔV sMrab;bnÞúk P Edlcl½tBITItaMg x1 eTA x2 enAelIFñwmedayKuN P CamYynwgbMEr


bMrYlGredaenrbs;ExST§iBl Edl ¬ y2 − y1 ¦. RbsinebICMralrbs;ExST§iBlKW s enaH ( y2 − y1 ) =
s ( x2 − x1 ) dUcenH

ΔV = Ps(x2 − x1 ) sMrab;ExST§iBlEdlmanCMral (6-1)

RbsinebIbnÞúkcl½tqøgcMnucEdlExST§iBlmanPaBdac; dUcbnÞúk C enAkñúgrUbTI 6-27a enaHbMEr
bMrYlkMlaMgkat;manlkçN³samBaØbMputKW
ΔV = P( y 2 − y1 ) sMrab;ExST§iBlEdlmanPaBdac; (6-2)

kareRbIsmIkarxagelICamYynwgFñwmrgbnÞúkbgðajBIExST§iBlsMrab; VC ¬rUbTI 6-27a¦. cMNaMfatMél
rbs;CMralénExST§iBlKW s = 0.75 /(12 − 3) = 0.75 / 9 = 0.0833 ehIyPaBdac;enARtg;cMnuc C manTMhM
0.75 + 0.25 = 1 . BicarNabnÞúkénkrNITI1 Edlcl½t 1.5m eTAkrNITI2 ¬rUbTI 6-27b¦. enAeBl

ekItmankrNIenH bnÞúk 4.5kN Føak;cuH − 1 ehIybnÞúkTaMgGs;cl½teLIgtamCMralrbs;ExST§iBl. dUc
enHbMErbMrYlkMlaMgkat;sMrab;krNIenHKW
ΔV1− 2 = 4.5(− 1) + [4.5 + 18 + 18](0.0833)(1.5) = +0.563kN
edaysar ΔV1− 2 viC¢man enaHkrNITI2nwgmanVC FMCagkrNITI1. eRbobeFobcMelIy (VC )1 nig (VC )2
Edl)anKNnaBImun EdlCak;Esþg (VC )2 = (VC )1 + 0.563 . eFVIkarGegát ΔV2−3 EdlekItmanenAeBl
krNITI2cl½teTAkrNITI3 ¬rUbTI 6-27b¦ eyIgRtUvKitkarFøak;cuHénbnÞúk18kN nigkarcl½ttamTis
edk1.5m énbnÞúkTaMgGs;elICMralrbs;ExST§iBl. eyIg)an
ΔV2 − 3 = 18(− 1) + [4.5 + 18 + 18](0.0833)(1.5) = −12.94kN
edaysar ΔV2−3 GviC¢man enaHkrNITI2CaTItaMgénbnÞúkeRKaHfñak; dUckarkMNt;elIkmun.
m:Um:g;Bt;³ eyIgk¾GaceRbIviFIelIkmunedIm,IkMNt;TItaMgeRKaHfñak;énes‘rIénbnÞúkcMcMnucEdlbegáItm:Um:g;Bt;FM
bMputenARtg;TItaMgCak;lak;enAelIeRKOgbgÁúM. dMbUg eKcaM)ac;RtUvKUrExST§iBlsMrab;m:Um:g;enARtg;cMnuc
nigkMNt;CMral s rbs;kMNat;ExST§iBlrbs;va. sMrab;clnatamTisedk (x2 − x1 ) rbs;bnÞúkcMcMnuc P
bMErbMrUlm:Um:g;Bt; ΔM smmUleTAnwgTMhMrbs;bnÞúkKuNnwgbMErbMrYlGredaenrbs;ExST§iBleRkamGMeBI
rbs;bnÞúk.
ΔM = Ps ( x2 − x1 ) sMrab;ExST§iBlEdlmanCMral (6-3)

Ca]TahrN_ BicarNaFñwm bnÞúk nigExST§iBlsMrab;m:Um:g;enARtg;cMnuc C enAkñúgrUbTI 6-28a. RbsinebI
eKdak;bnÞúknImYy²énbnÞúkcMcMnucTaMgbIenAelIFñwmeGayRtUvKñanwgkMBUlrbs;ExST§iBl eyIgnwgTTYl)an
T§iBlFMbMputBIkMlaMgnImYy². krNIénkardak;bnÞúkbIRtUv)anbgðajenAkñúgrUbTI 6-28b. enAeBlbnÞúk
énkrNITI1 cl½t)an1.2m eTAeqVg eyIgnwg)ankrNITI2. eyIgeXIjfa bnÞúk 9kN kat;bnßy ΔM1− 2


edaysarCMral (2.25 / 3)manTisedAcuHeRkam ¬rUbTI 6-28a¦. dUcKña kMlaMg18kN nig13.5kN eFVI
eGay ΔM 1− 2 ekIneLIg edaysarCMral [2.25 /(12 − 3)] manTisedAeLIgelI. eyIgman
⎛ 2.25 ⎞ ⎛ 2.25 ⎞
ΔM 1− 2 = −9⎜ ⎟(1.2 ) + (18 + 13.5)⎜ ⎟(1.2) = 1.35kN .m
⎝ 3 ⎠ ⎝ 12 − 3 ⎠
edaysar ΔM 1− 2 viC¢man eyIgRtUveFVIkarGegátbnþeTotedaycl½tbnÞúk1.8m BIkrNITI2 eTAkrNITI3.
⎛ 2.25 ⎞ ⎛ 2.25 ⎞
ΔM 2 − 3 = −(9 + 18)⎜ ⎟(1.8) + 13.5⎜ ⎟(1.8) = −30.38kN .m
⎝ 3 ⎠ ⎝ 12 − 3 ⎠
enATIenH bMErbMrYlGviC¢man dUcenHm:Um:g;Bt;FMbMputenARtg;cMnuc C nwgekItmanenAeBlFñwmrgbnÞúkdUc
bgðajenAkñúgkrNITI2 ¬rUbTI 6-28c¦. dUcenHm:Um:g;GtibrmaenARtg; C KW
(M C )max = 9(1.35) + 18(2.25) + 13.5(1.8) = 77.0kN .m
]TahrN_xageRkamnwgbgðajBIbEnßmBIviFIenH.

]TahrN_ 6-18³ kMNt;kMlaMgkat;viC¢manGtibrmaEdlekItmanenARtg;cMnuc B enAkñúgFñwmEdlbgðaj
enAkñúgrUbTI 6-29a EdlbNþalBIbnÞúkkg;rbs;rfynþdwkTMnij.
dMeNaHRsay³
ExST§iBlsMrab;kMlaMgkat;enARtg;cMnuc B RtUv)anbgðajenAkñúgrUbTI 6-29b.
karcl½t 0.9m rbs;bnÞúk18kN ³ KitfabnÞúk18kN eFVIGMeBIenAxagsþaMcMnuc B bnþicEdleFIVeGayeyIg
TTYl)anT§iBlviC¢manGtibrma. edaysarkMNat;Fñwm BC manRbEvg 3m enaHbnÞúk 4.5kN minTan;man
GMeBIelIFñwmeT. enAeBlrfynþdwkTMnijcl½t 0.9m eTAeqVg bnÞúk18kN Føak;cuHenAelIExST§iBl1Éktþa



ehIybnÞúk18kN / 40.5kN nig 67.5kN eFVIeGay ΔVB viC¢manekIneLIg edaysarCMralekIneLIgeTAeqVg.
eTaHbICabnÞúk 45kN cl½teTAmux 0.9m k¾eday k¾vaenAminTan;manGMeBIelIFñwmEdr. dUcenH
⎛ 0.5 ⎞
ΔVB = 18(− 1) + (18 + 40.5 + 67.5)⎜ ⎟0.9 = +0.9kN
⎝ 3 ⎠

karcl½t1.8m rbs;bnÞúk 40.5kN ³ enAeBlbnÞúk 40.5kN manGMeBIenAxagsþaMcMnuc B bnþic ehIybnÞab;
mkrfynþdwkTMnijcl½t1.8m eTAeqVg eyIgman
⎛ 0.5 ⎞ ⎛ 0.5 ⎞
ΔVB = 40.5(− 1) + (18 + 40.5 + 67.5)⎜ ⎟(1.8) + 4.5⎜ ⎟(1.2) = +6.3kN
⎝ 3 ⎠ ⎝ 3 ⎠
cMNaMfakñúgkarKNnabnÞúk 45kN cl½t)an1.2m enAelIFñwm.
karcl½t1.8m rbs;bnÞúk 67.5kN ³ RbsinebIbnÞúk 67.5kN manTItaMgenAxagsþaMcMnuc B bnþic ehIybnÞab;
mkrfynþdwkTMnijcl½teTAeqVg)ancMgay1.8m enaHbnÞúk18kN enAsl;EtcMgay 0.3m eTot vanwgecj
BIFñwm dUcKñaRbsinbnÞúk 40.5kN cl½tEfm1.2m vak¾nwgcakecjBIFñwmEdr. dUcenH
⎛ 0.5 ⎞ ⎛ 0.5 ⎞ ⎛ 0.5 ⎞
ΔVB = 67.5(− 1) + 18⎜ ⎟(0.3) + 40.5⎜ ⎟(1.2) + (67.5 + 45)⎜ ⎟(1.8) = −24.8kN
⎝ 3 ⎠ ⎝ 3 ⎠ ⎝ 3 ⎠
edaysar ΔVB GviC¢man TItaMgd¾RtwmRtUvrbs;bnÞúkekItmanenAeBlbnÞúk 67.5kN sßitenAxagsþaMcMnuc B
bnþic ¬rUbTI 6-29c¦. dUcenH
(VB )max = 18(− 0.05) + 40.5(− 0.2) + 37.5(0.5) + 45(0.2) = 33.8kN
kñúgkarGnuvtþ eKk¾GacBicarNaclnarbs;rfynþdwgTMnijBIeqVgeTAsþaM ehIybnÞab;mkeRCIserIstMél
GtibrmarvagsßanPaBTaMgBIrenH.



]TahrN_ 6-19³ kMNt;m:Um:g;Bt;viC¢manGtibrmaEdlekItmanenARtg;cMnuc B enAkñúgFñwmEdlbgðaj
enAkñúgrUbTI 6-30a EdlbNþalBIbnÞúkkg;rbs;]bkrN_eyag (crane).

dMeNaHRsay³
ExST§iBlsMrab;m:Um:g;Bt;enARtg;cMnuc B RtUv)anbgðajenAkñúgrUbTI 6-30b.
karcl½t 2m rbs;bnÞúk 3kN ³ RbsinebIeKsnμt;fabnÞúk 3kN eFVIGMeBIenARtg;cMnuc B ehIybnÞab;mkva
cl½t 2m eTAxagsþaM ¬rUbTI 6-30b¦ enaHbMErbMrYlm:Um:g;Bt;KW
⎛ 1.20 ⎞ ⎛ 1.20 ⎞
ΔM B = −3⎜ ⎟(2) + 8⎜ ⎟(2) = 7.20kN .m
⎝ 3 ⎠ ⎝ 2 ⎠
ehtuGIVeKminKitbBa©ÚlbnÞúk 4kN kñúgkarKNna?
karcl½t 3m rbs;bnÞúk 8kN ³ RbsinebIeKsnμt;fabnÞúk 8kN eFVIGMeBIenARtg;cMnuc B ehIybnÞab;mkva
cl½t 3m eTAxagsþaM enaHbMErbMrYlm:Um:g;Bt;KW
⎛ 1.20 ⎞ ⎛ 1.20 ⎞ ⎛ 1.20 ⎞
ΔM B = −3⎜ ⎟(3) − 8⎜ ⎟(3) + 4⎜ ⎟(2) = −8.40kN .m
⎝ 3 ⎠ ⎝ 2 ⎠ ⎝ 2 ⎠
cMNaMfa enATIenHbnÞúk 4kN sßitenAxageRkAFñwm1m dUcenHvacl½telIFñwmEt 2m b:ueNÑaH.
edaysar ΔM B pøas;bþÚrsBaØa TItaMgBitR)akdrbs;bnÞúksMrab;m:Um:g;viC¢manGtibrmaenARtg; B
ekItmanenAeBlEdlkMlaMg 8kN sßitenARtg; B ¬rUbTI 6-30b¦. dUcenH
(M B )max = 8(1.20) + 3(0.4) = 10.8kN .m

]TahrN_ 6-20³ kMNt;kMlaMgsgát;GtibrmaEdlekItmanenAkñúgGgát; BG én truss xagenAkñúgrUbTI
6-31a EdlbNþalBIbnÞúkkg;xagsþaMrbs;rfynþ nigrfseNþag. snμt;fabnÞúkGnuvtþedaypÞal;eTAelI
truss nigcl½teTAEtsþaM.



dMeNaHRsay³
ExST§iBlsMrab;kMlaMgenAkñúgGgát; BG RtUv)anbgðajenAkñúgrUbTI 6-31b. enATIenH eyIgnwgeRbIviFI
trial-and error sMrab;karedaHRsaycMeNaTenH. edaysareyIgcg;)ankMlaMgGviC¢man ¬sgát;¦FMbMput

enAkñúgGgát; BG eyIgcab;epþImedaHRsaydUcxageRkam³
bnÞúk1.5kN enARtg;cMnuc C ³ enAkñúgkrNIenH
⎛ 0.3125 ⎞
FBG = 1.5kN (− 0.625) + 4(0) + 2kN ⎜ ⎟(1m ) = −0.729kN
⎝ 3m ⎠
bnÞúk 4kN enARtg;cMnuc C ³ edayGegátkrNIenH eyIgeXIjfavamanlkçN³smRsbCagkrNIelIkmun
⎛ − 0.625 ⎞
FBG = 4kN (− 0.625) + 1.5kN ⎜ ⎟(4m ) + 2kN (0.3125) = −2.50kN
⎝ 6m ⎠
bnÞúk 2kN enARtg;cMnuc C ³ enAkñúgkrNIenHbnÞúkTaMgGs;nwgbegáItkMlaMggsgát;enAkñúgGgát; BC .
⎛ − 0.625 ⎞ ⎛ − 0.625 ⎞
FBC = 2kN (− 0.625) + 4kN ⎜ ⎟(3m ) + 1.5kN ⎜ ⎟(1m ) = −2.66kN
⎝ 6m ⎠ ⎝ 6m ⎠
edaysarlT§plénkrNIcugeRkaypþl;nUvcMelIyFMbMput bnÞúkeRKaHfñak;ekItmanenAeBlbnÞúk 2kN enA
Rtg;cMnuc C .

^>&> kMlaMgkat; nigm:Um:g;Gtibrmadac;xat (Absolute maximum shear and moment)
enAkñúgkfaxNÐ 6-6 eyIg)anbegáItviFIsMrab;KNnakMlaMgkat; nigm:Um:g;Bt;GtibrmaenARtg;cMnuc
Cak;lak;enAelIFñwmEdlbNþalBIes‘rIénbnÞúkcl½tcMcMnuc. cMeNaTEdlmanlkçN³TUeTACagEdlBak;B½n§
nwgkarkMNt;TItaMgrbs;cMnucenAelIFñwm nigTItaMgrbs;bnÞúkenAelIFñwmEdlGaceGayeKTTYl)ankMlaMg
kat;Gtibrmadac;xat nigm:Um:g;Bt;Gtibrmadac;xatEdlekItBIbnÞúkxageRkA. RbsinebIFñwmCaFñwmkugs‘ul
b¤FñwmTMrsamBaØ eKGacedaHRsaydUcxageRkam³



kMlaMgkat;³ sMrab;Fñwmkugs‘ulkMlaMgkat;Gtibrmadac;xatnwgekItmanenARtg;cMnucEdlmanTItaMgEk,r
TMrbgáb;. eKGacrkkMlaMgkat;GtibrmaedayviFImuxkat; CamYynwgbnÞúkEdlmanTItaMgNamYyenAelIElVg
Fñwm ¬rUbTI 6-32¦.

sMrab;FñwmTMrsamBaØkMlaMgkat;Gtibrmadac;xatnwgekItmanenAEk,rTMrNamYy. enAkñúgkrNIenH
eKdak;bnÞúky:agNaeGaybnÞúkmYyenAEk,rTMrCabnþbnÞab; dUcenAkñúgrUbTI 6-33.

m:Um:g;Bt;³ m:Um:g;Bt;Gtibrmadac;xatsMrab;Fñwmkugs‘ulekItmanenARtg;cMnucdUcKñanwgkarekItmankMlaMg
kat;Gtibrmadac;xat eTaHbICaenAkñúgkrNIenHbnÞúkcMcMnucmanTItaMgenAcugq¶ayBITMrk¾eday dUcenAkñúg
rUbTI 6-34.

CaFmμta sMrab;FñwmTMrsamBaØeKminGackMNt;TItaMgeRKaHfñak;rbs;bnÞúk nigm:Um:g;Gtibrmadac;xat
edaykarGegát)aneT. b:uEnþeyIgGackMNt;TItaMgedaykarviPaK. ]bmaeKmanFñwmrgkMlaMg F1 / F2 /
F3 EdlbgðajenAkñúgrUbTI 6-35a. edaysardüaRkamm:Um:g;sMrab;es‘rIénkMlaMgcMcMnucpSMeLIgeday

kMNat;bnÞat;Rtg;EdlmankMBUlenARtg;kMlaMgnImYy² m:Um:g;Gtibrmadac;xatnwgekItmaneRkamGMeBIén
kMlaMgmYyénkMlaMgTaMgbI. snμt;fam:Um:g;GtibrmaenHekItmaneRkamGMeBI F2 . TItaMgénbnÞúk F1 / F2 /
F3 enAelIFñwmRtUv)ankMNt;edaycMgay x Edlvas;BI F2 eTAcMnuckNþalrbs;FñwmdUcbgðaj. edIm,IkMNt;

tMélCak;lak;rbs; x dMbUgeyIgRtUvkMNt;kMlaMgpÁÜbrbs;RbB½n§kMlaMg FR nigcMgay x' Edlvas;BI F2 .
bnÞab;mkeTot eFVIplbUkm:Um:g;Rtg;cMnuc B eyIgnwg)ankMlaMgRbtikmμ Ay xageqVgrbs;Fñwm Edl
∑MB = 0 Ay =
1
⎢2 ( )
(FR )⎡ L − x'− x ⎤
⎥
L ⎣ ⎦



RbsinebIeKkat;FñwmenAxageqVg F2 bnþic enaHdüaRkamGgÁesrIrbs;vaRtUv)anbgðajenAkñúgrUbTI 6-35b.
dUcenH m:Um:g; M 2 eRkamGMeBI F2 KW
⎛L ⎞
∑M = 0 M 2 = Ay ⎜ − x ⎟ − F1d1
⎝2 ⎠
⎡L
( )
⎤⎛ L ⎞
= (FR )⎢ − x'− x ⎥⎜ − x ⎟ − F1d1
1
L ⎣2 ⎦⎝ 2 ⎠
2
F L F x' F x F x x'
= R − R − R + R − F1d1
4 2 2 L
edIm,IeGay M 2 Gtibrma eKTamTareGay
dM 2 2 F x F x'
=− R + R =0
dx L L
b¤ x=
x'
2
dUcenH eyIgGacsnñidæanfam:Um:g;Gtibrmadac;xatenAkñúgFñwmTMrsamBaØekItmaneRkamGMeBIénkMlaMgcMcMnuc
mYy EdlkMlaMgenHmanTItaMgenAelIFñwmEbbNaEdlva nigkMlaMgpÁÜbénRbB½n§kMlaMgmancMgayesμIKñaBI
cMnuckNþalFñwm. edaysareKmanes‘rIénkMlaMgenAelIElVg ¬]TahrN_ F1 / F2 / F3 kñúgrUbTI 6-35a¦
eKRtUvGnuvtþeKalkarN_eTAelIbnÞúknImYy²énes‘rIbnÞúk nigKNnam:Um:g;GtibrmaEdlRtUvnwgkMlaMgenaH.
edaykareRbobeFob m:Um:g;EdlmanTMhMFMCageKCam:Um:g;Gtibrmadac;xat. CaTUeTA m:Um:g;Gtibrmadac;
xateRcInekItmaneRkamGMeBIkMlaMgEdlFMCageKEdlsßitenAEk,rkMlaMgpÁÜbrbs;RbB½n§kMlaMg.

kBa©b;éntMélExST§iBlGtibrma³ c,ab; b¤rUbmnþsMrab;kMNt;kMlaMgkat; b¤m:Um:g;Gtibrmadac;xatmankar
Bi)aksMrab;FñwmEdlxusBIFñwmkugs‘ul nigFñwmTMrsamBaØEdl)anerobrab;enATIenH. b:uEnþ edIm,IedaHRsay
cMeNaTEbbenH eKRtUvsg;ExST§iBlsMrab;kMlaMgkat; nigm:Um:g;Bt;enARtg;cMnuceRCIserIstambeNþayFñwm
ehIybnÞab;mkKNnakMlaMgkat; nigm:Um:g;Bt;sMrab;cMnucnImYy²edayeRbIviFIenAkñúgkfaxNÐ 6-6. eRkay
eBlsg;düaRkaméntMélTaMgenH eKnwg)ankBa©b;éntMélGtibrmaBItMélkMlaMgkat; b¤m:Um:g;Bt;Gtibrma
dac;xat ehIyeKGackMNt;TItaMgrbs;va. CakarBit eKENnaMeGayeRbIkmμviFIkMuBüÚT½rsMrab;edaHRsay


cMeNaTsμúKsμaj edaysarkarviPaKedayédmanlkçN³saMjúaM.

]TahrN_ 6-21³ kMNt;m:Um:g;Gtibrmadac;xatenAkñúgkMrals<anTMrsamBaØEdlbgðajenAkñúgrUbTI 6-
36a.

dMeNaHRsay³
dMbUgeKRtUvkMNt;TMhM nigTItaMgrbs;kMlaMgpÁÜbénRbB½n§ ¬rUbTI 6-36a¦. eyIgman
+ ↓ FR = ∑ F FR = 8 + 6 + 4 = 18kN
+ M RC = ∑ M C 18 x = 6(3) + 4(4.5) x = 2m

dMbUgeyIgsnμt;fam:Um:g;Gtibrmadac;xatekItmaneRkamGMeBIénbnÞúk 6kN . eKdak;bnÞúk nigbnÞúkpÁÜbenA
cMgayesμIKñaBIcMnuckNþalrbs;Fñwm ¬rUbTI 6-36b¦. dMbUg KNna Ay ¬rUbTI 6-36b¦ eyIgman
+ ∑MB = 0 − Ay (9) + 18(5) = 0 Ay = 10kN

edayeRbImuxkat;xageqVgrbs;Fñwm ¬rUbTI 6-36c¦ eyIg)an
+ MS = 0 − 10(5) + 8(3) + M S = 0 M S = 26.0kN .m
m:Um:g;Gtibrmadac;xatGacmanlT§PaBekItmaneRkamGMeBIénbnÞúk 8kN edaysar 8kN > 6kN ehIy
edaysar FR sßitenAcenøaHbnÞúk 8kN nig 6kN . edIm,IGegátkrNIenH eKRtUvdak;bnÞúk 8kN nig FR



cMgayesμIKñaBIcMnuckNþalrbs;Fñwm ¬rUbTI 6-36d¦. bgðajfa Ay = 7kN dUcbgðajenAkñúgrUbTI 6-
36e nigbgðajfa
M S = 24.5kN .m
edaykareRbobeFob m:Um:g;Gtibrmadac;xagKW
M S = 26kN .m
EdlekItmaneRkamGMeBIénbnÞúk 6kN enAeBlEdlbnÞúkmanTItaMgenAelIFñwmdUcbgðajenAkñúgrUbTI 6-
36b.



cMeNaT
6>1> KUrExST§iBlsMrab; (a) kMlaMgRbtikmμbBaÄr 6>6> edaHRsaycMeNaT 6>5> edayeRbIeKal-
enARtg;cMnuc A (b) kMlaMgkat;enARtg;cMnuc C nig karN_muyeyeRbsøÚ.
(c) m:Um:g;Bt;enARtg;cMnuc C . edaHRsaycMeNaT 6>7> KUrExST§iBlsMrab; (a) kMlaMgkat;enARtg;
enHedayeRbIviFImUldæanénkfaxNÐ 6-1. TMrbgáb; A nig (c) m:Um:g;Bt;enARtg;cMnuc B .

6>8> edaHRsaycMeNaT 6>7> edayeRbIeKal-
6>2> edaHRsaycMeNaT 6>1> edayeRbIeKal- karN_muyeyeRbsøÚ.
karN_muyeyeRbsøÚ. 6>9> KUrExST§iBlsMrab; (a) kMlaMgRbtikmμbBaÄr
6>3> KUrExST§iBlsMrab; (a) kMlaMgRbtikmμbBaÄr enARtg;cMnuc B (b) m:Um:g;Bt;enARtg;cMnuc C nig
enARtg;cMnuc A (b) m:Um:g;Bt;enARtg;cMnuc A nig (c) kMlaMgkat;enARtg;cMnuc C . snμt;fa A CaTMr
(c) kMlaMgkat;enARtg;cMnuc B . snμt;faTMrenA snøak; ehIy B CaTMrkl;. edaHRsaycMeNaT
Rtg;cMnuc A CaTMrbgáb;. edaHRsaycMeNaT enHedayeRbIviFImUldæanénkfaxNÐ 6-1.
enHedayeRbIviFImUldæanénkfaxNÐ 6-1.

6>4> edaHRsaycMeNaT 6>3> edayeRbIeKal- 6>10> edaHRsaycMeNaT 6>9> edayeRbIeKal-
karN_muyeyeRbsøÚ. karN_muyeyeRbsøÚ.
6>5> KUrExST§iBlsMrab; (a) kMlaMgRbtikmμbBaÄr 6>11> KUrExST§iBlsMrab; (a) kMlaMgRbtikmμ
enARtg;cMnuc A (b) kMlaMgkat;enARtg;cMnuc C nig bBaÄrenARtg;cMnuc B (b) kMlaMgkat;enARtg;
(c) m:Um:g;Bt;enARtg;cMnuc C . edaHRsaycMeNaT
cMnuc C nig (c) m:Um:g;Bt;enARtg;cMnuc C . edaH
enHedayeRbIviFImUldæanénkfaxNÐ 6-1. RsaycMeNaTenHedayeRbIviFImUldæanénkfaxNÐ
6-1.

Problems T.Chhay -203


elxsMrab;kMBUl. FñwmenHmansnøak;enARtg;TMr B
ehIy A CaTMrsnøak;. edaHRsaycMeNaTenH
edayeRbIviFI mUldæanénkfaxNÐ 6-1.
6>12> edaHRsaycMeNaT 6>10> edayeRbI
eKalkarN_muyeyeRbsøÚ.
6>13> KUrExST§iBlsMrab; (a) m:Um:g;Bt;enARtg;
cMnuc A nig(b) m:Um:g;Bt;enARtg;cMnuc B . snμt;fa
A CaTMrbgáb;. edaHRsaycMeNaT enHedayeRbI 6>18> edaHRsaycMeNaT 6>17> edayeRbI
viFImUldæanénkfaxNÐ 6-1. eKalkarN_muyeyeRbsøÚ.
6>19> FñwmmYyRTbnÞúkefrBRgayesμI 500 N / m
nigkMlaMgcMcMnucGefreTal 3000 N . kMNt; (a)
m:Um:g;viC¢manGtibrmaEdlGacekItmanenARtg;
6>14> edaHRsaycMeNaT 6>13> edayeRbI cMnuc C nig(b) kMlaMgkat;viC¢manGtibrmaEdl
eKalkarN_muyeyeRbsøÚ. GacekItmanenARtg;cMnuc C . snμt; A CaTMrsnøak;
6>15> KUrExST§iBlsMrab; (a) m:Um:g;Bt;enARtg; nig B CaTMrkl;.
cMnuc C (b) kMlaMgRbtikmμbBaÄrenARtg;cMnuc A
nig (c) kMlaMgkat;bBaÄrenARtg;cMnuc B . cMnuc E
Casnøak;. edaHRsaycMeNaTenHedayeRbIviFI
mUldæanénkfaxNÐ 6-1.
6>20> bnÞúkGefrBRgayesμI10.5kN / m nigbnÞúk
cMnucGefreTal 45kN RtUv)andak;elIFñwm. kMNt;
(a) kMlaMgRbtikmμbBaÄrGefrviC¢manGtibrmaenA
6>16> edaHRsaycMeNaT 6>15> edayeRbI Rtg;TMr A (b) kMlaMgkat;GefrviC¢manGtibrmaenA
eKalkarN_muyeyeRbsøÚ. Rtg;cMnuc C (c) m:Um:g;GefrviC¢manGtibrmaenA
6>17> KUrExST§iBlsMrab; (a) kMlaMgkat;enARtg; Rtg;cMnuc C . snμt; A CaTMrkl; nig B CaTMrsnøak;.
cMnuc C (b) m:Um:g;Bt;enARtg;cMnuc C nig (c)
kMlaMgkat;bBaÄrenARtg;cMnuc D . bgðajtMélCa
cMeNaT T.Chhay -204


6>21> KUrExST§iBlsMrab; (a) kMlaMgRbtikmμ 6>25> bnÞúkGefrBRgayesμI 7.5kN / m nig
bBaÄrenARtg;cMnuc B (b) m:Um:g;Bt;enARtg;cMnuc E . kMlaMgcMcMnuceTal 90kN manGMeBIenAelIFñwm.
snμt;TMr B nig D CaTMrkl; (roller). enARtg; kMNt; (a) kMlaMgRbtikmμbBaÄrviC¢manGtibrma
cMnuc C Casnøak;. edaHRsaycMeNaTenHeday enARtg;TMr B (b) kMlaMgkat;GrefrviC¢manGti-
eRbIviFImUldæanénkfaxNÐ 6-1. brmaenARtg;cMnuc C nig (c) m:Um:g;GefrviC¢man
GtibrmaenARtg;cMnuc C . snμt;fa A CaTMrsnøak;
nig B CaTMrkl;.

6>22> edaHRsaycMeNaT 6>21> edayeRbI
eKalkarN_muyeyeRbsøÚ.
6>23> FñwmsmasrgbnÞúkefrBRgayesμI 2kN / m 6>26> KUrExST§iBlsMrab;kMlaMgTajkñúgkab
BC . kMras;rbs;FñwmRtUv)anecal. etIkMlaMg
nigbnÞúkGefrBRgayesμI1.5kN / m . kMNt; (a)
m:Um:g;Bt;GviC¢manGtibrmaEdlekItenARtg;cMnuc TajGtibrmakñúgkabEdlbNþalBIbnÞúk 30kN
A nig (b) kMlaMgkat;viC¢manGtibrmaenARtg;
manTMhMb:unμan?
cMnuc D . snμt; B CatMNsnøak; nig C CaTMrkl;.

6>24> FñwmrgbnÞúkefrBRgayesμI 500 N / m nig
kMlaMgcMcMnucGefreTal 3000 N . kMNt; (a)
m:Um:g;Bt;GefrGtibrmaEdlGacekItmanenARtg;cM 6>27> FñwmrgbnÞúkGefrBRgayesμI 0.6kN / m nig
nuc C nig (b) kMlaMgkat;viC¢manGtibrmaEdl kMlaMgcMcMnucGefr 0.8kN . kMNt; (a) m:Um:g;Bt;
GacekItmanenARtg;cMnuc C . snμt; B CaTMr GefrviC¢manGtibrmaEdlGacekItmanenARtg;
snøak; nig A CaTMrkl;. cMnuc B nig (b) kMlaMgRbtikmμbBaÄr enARtg;


cMnuc D . snμt; C CaTMrsnøak; nig D CaTMrkl;. kat;GefrFMbMputenARtg;cMnuc D ? kMNt;tMélén
m:Um:g;Bt; nigkMlaMgkat;. snμt; A CaTMrbgáb; B Ca
tMNsnøak; nig C CaTMrkl;.
6>31> KUrExST§iBlsMrab; (a) kMlaMgkat;kñúg
panel BC rbs;r:t (b) m:Um:g;Bt;enARtg;cMnuc A .
6>28> FñwmsmasrgbnÞúkefrBRgayesμI snμt; B CaTMrkl; nig D CaTMrsnøak;.
1.5kN / m nig kMlaMgcMcMnucefreTal 10kN .

kMNt; (a) m:Um:g; Bt; GviC¢manGtibrmaEdlGac
ekItmanenARtg; cMnuc A nig (b) kMlaMgkat;
viC¢manenARtg;cMnuc B . snμt; A CaTMrbgáb; B Ca 6>32> bnÞúkBRgayGefr 7.5kN / m nigbnÞúk
tMNsnøak; nig C CaTMrkl;. cMcMnucGefr 9kN manGMeBIenAelIkMralxNÐ.
kMNt; (a) kMlaMgkat;GefrviC¢manGtibrmaenAkñúg
panel DE nig (b) m:Um:g;Bt;GefrGviC¢manGti-

brmaenARtg;cMnuc C .

6>29> eKRtUvdak;bnÞúkGefreTal 2kN enAelIFñwm
Rtg;TItaMgNaedIm,IeGayvabegáItm:Um:g;Bt;GefrFM
bMputenARtg;cMnuc D ? etIm:Um:g;Bt;enHmanTMhM 6>33> KUrExST§iBlsMrab; (a) kMlaMgkat;kñúg
b:unμan? snμt; A CaTMrbgáb; B Ca tMNsnøak; panel DE rbs;r:t (b) m:Um:g;Bt;enARtg;cMnuc C .

nig C CaTMrkl;.

panel CD rbs;r:t (b) m:Um:g;Bt;enARtg;cMnuc E .
6>30> etIeKRtUvdak;bnÞúkGefrBRgayesμI
6kN / m Rtg;kEnøgNaenAelIFñwm ABC én

cMeNaT 6>29 edIm,I TTYl)an (a) m:Um:g;Bt;
GrefrFMbMputenARtg;cMnuc A nig (b) kMlaMg
cMeNaT T.Chhay -206


6>35> bnÞúkBRgayGefr 6kN / m nigbnÞúkGrfr snμt;fa D CaTMrkl; nig G CaTMrsnøak;.
cMcMnuc 9kN manGMeBIelIkMralxNÐ. kMNt; (a)
kMlaMgkat;GefrGtibrmaenAkñúg panel CD nig
(b) m:Um:g;GefrGtibrmaenARtg;cMnuc B .

6>39> KUrExST§iBlsMrab; (a) m:Um:g;Bt;enAkñúgr:t
Rtg;cMnuc D (b) kMlaMgkat;kñúg panel BC rbs;
r:t.
6>36> bnÞúkBRgayGefr 27kN / m nigbnÞúk
GrfrcMcMnuc 60kN manGMeBIelIkMralxagelIFñwm.
RbsinebIFñwm k¾RTbnÞúkBRgayefr 5.25kN / m Edr
enaH cUr kMNt; (a) kMlaMgkat;GtibrmaenAkñúg 6>40> KUrExST§iBlsMrab;m:Um:g;Bt;enAkñúgr:tRtg;
panel BC rbs;r:t nig (b) m:Um:g;Bt;Gtibrma cMnuc B . kMNt;m:Um:g;Bt;GefrviC¢manGtibrmaenA
enAkñúg r:tRtg;cMnuc C . kñúgr:tRtg;cMnuc B RbsinebIkMlaMgGefrcMcMnuc
eTal 50kN cl½telIkMralxagelIFñwm. snμt;TMr
rbs;FñwmTaMg enHGacGnuvtþkMlaMgEdlmanTis
eLIgelI nig kMlaMgEdlmanTiscuHeRkam
eTAelIFñwm.
panel BC rbs;r:t (b) m:Um:g;Bt;enARtg;cMnuc C .

snμt; B CaTMrsnøak; nig D CaTMrkl;.
6>41> KUrExST§iBlsMrab;m:Um:g;Bt;enAkñúgr:tRtg;
cMnuc F . kMNt;m:Um:g;Bt;GefrviC¢manGtibrmaenA
kñúgr:tRtg;cMnuc F RbsinebIkMlaMgGefrcMcMnuc
6>38> bnÞúkBRgayGefr 5kN / m nigbnÞúkGrfr eTal 8kN cl½telIkMralxagelI. snμt;TMrrbs;
cMcMnuc18kN manGMeBIelIkMral. kMNt; (a) FñwmTaMg enHGacGnuvtþkMlaMgEdlmanTiseLIgelI
kMlaMgRbtikmμbBaÄrenARtg;cMnuc D (b) kMlaMg nig kMlaMgEdlmanTiscuHeRkameTAelIFñwm.
kat;viC¢manGtibrmaenAkñúg panel EF rbs;r:t
nig (c) m:Um:g;Bt;viC¢manGtibrmaenARtg;cMnuc F .


kMBUlrbs;ExSekag. Ggát;TaMgGs;manRbEvgesμI
Kña.

6>42> KUrExST§iBlsMrab;kMlaMgkat;enAkñúg
panel CD rbs;r:t. kMNt;kMlaMgkat;Gefr

GviC¢manGtibrmaenAkñúg panel CD Edl
bNþalBIbnÞúkBRgayGefr 10kN / m Edlman 6>46> KUrExST§iBlsMrab;kMlaMgenAkñúgGgát; FC
GMeBIenAelIkMralxagelIFñwm. én Warren truss éncMeNaT 6>45 . bgðaj
tMélCaelxsMrab;kMBUlrbs;ExSekag. Ggát;TaMg
Gs;manRbEvgesμIKña.
6>47> KUrExST§iBlsMrab;kMlaMgenAkñúgGgát;
6>43> KUrExST§iBlsMrab;m:Um:g;Bt;enARtg;cMnuc DE én Warren truss éncMeNaT 6>45 .
E rbs;r:téncMeNaT 6>41. kMNt;m:Um:g;Bt; bgðaj tMélCaelxsMrab;kMBUlrbs;ExSekag.
GefrviC¢manGtibrmaenAkñúgr:tRtg;cMnuc E Rbsin Ggát;TaMgGs;manRbEvgesμIKña.
ebIkMlaMgGefrcMcMnuc 40kN cl½tenAelIkMral 6>48> KUrExST§iBlsMrab;kMlaMgenAkñúg (a) Ggát;
xagelIFñwm. EH nig (b) Ggát; JE .
6>44> bnÞúkBRgayGefr 5kN / m nigbnÞúkGrfr
cMcMnuc18kN manGMeBIelIkMral. kMNt; (a)
kMlaMgRbtikmμbBaÄrenARtg;cMnuc D (b) kMlaMg
kat;GefrGtibrmaenAkñúg panel DE rbs;r:t nig 6>49> KUrExST§iBlsMrab;kMlaMgenAkñúgGgát; JI
(c) m:Um:g;Bt;viC¢manGtibrmaenARtg;cMnuc E . én truss kñúgcMeNaT 6>48.
snμt; D CaTMrkl; nig G CaTMrsnøak;. 6>50> KUrExST§iBlsMrab;kMlaMgenAkñúgGgát; AL
én truss kñúgcMeNaT 6>48.
6>51> KUrExST§iBlsMrab;kMlaMgenAkñúgGgát; HG
bnÞab;mkkMNt;kMlaMgGefrGtibrma ¬kMlaMgTaj
6>45> KUrExST§iBlsMrab;kMlaMgenAkñúgGgát; b¤kMlaMgsgát;¦ EdlGacekItmanenAkñúgGgát;Edl
GF én Warren truss. bgðajtMélCaelxsMrab; bNþalBIbnÞúkBRgayGefr16kN / m Edlman
cMeNaT T.Chhay -208


GMeBIelIkMrals<antambeNþayGgát;xageRkam 6>57> eKeRbIdMbUl truss sMrab;BüÜrpøÚvén]brkN_
rbs; truss. eyagEdlP¢ab;eTAnwgGgát;xageRkamrbs; truss
dUcbgðaj. kMNt;kMlaMgGefrGtibrma ¬kMlaMg
Taj b¤kMlaMgsgát;¦ EdlGacekItmanenAkñúg
Ggát; GF EdlbNþalBIbnÞúk]bkrN_eyag
60kN . kMNt;TItaMg x rbs;bnÞúk. snμt; truss
6>52> KUrExST§iBlsMrab;kMlaMgenAkñúgGgát; HC RtUv)anRTedayTMrsnøak;Rtg;cMnuc A nigRtg;cMnuc
bnÞab;mkkMNt;kMlaMgGefrGtibrma ¬kMlaMgTaj E edayTMrkl;. ehIysnμt;faGgát;TaMgGs;RtUv
b¤kMlaMgsgát;¦ EdlGacekItmanenAkñúgGgát;Edl )antP¢ab;KñaRtg;tMNeday gusset plate.
bNþalBIbnÞúkBRgayGefr16kN / m Edlman
GMeBIelIkMrals<antambeNþayGgát;xageRkam
rbs; truss éncMeNaT 6>51.
AH bnÞab;mkkMNt;kMlaMgGefrGtibrma

¬kMlaMgTaj b¤kMlaMgsgát;¦ EdlGacekItman 6>58> kMNt;m:Um:g;Bt;GefrGtibrmaenARtg;cMnuc
enAkñúgGgát;Edl bNþalBIbnÞúkBRgayGefr A enAelIr:teTalEdlekIteLIgedaysarkUn
16kN / m Edlman GMeBIelIkMrals<antam reTHEdlmanm:as 2Mg ehIyTIRbCMuTMgn;enARtg;
beNþayGgát;xageRkam rbs; truss éncMeNaT G . snμt;fa A CaTMrkl;.
6>51.
RN én Baltimore truss.

6>59> rfynþdwkTMnijTMgn; 54kN GnuvtþkMlaMgkg;
dUcbgðajeTAelIr:ts<an. kMNt; (a) kMlaMgkat;
6>55> KUrExST§iBlsMrab;kMlaMgenAkñúgGgát; GefrFMbMputEdlekItmanenARtg;tMN C nig (b)
NG én truss kñúgcMeNaT 6>54.
m:Um:g;Bt;FMbMputenARtg;tMN C . snμt;farfynþcr
6>56> KUrExST§iBlsMrab;kMlaMgenAkñúgGgát; tamTisNak¾)antambeNþayG½kSkMral dUcenH
OC én truss kñúgcMeNaT 6>54.
bnÞúkrbs;vaRtUv)anepÞrBak;kNþaleTAr:txagTaMg
BIr. snμt;fatMN C CatMNbgáb; dUcenHvaGac


Tb;Tl;)anTaMgm:Um:g;Bt; nigTaMgkMlaMgkat;. kñúgr:txagRtg; C enAeBlEdlvaqøgkat;s<an.
snμt;farfynþcrtamTisNak¾)antambeNþay
G½kSkMral dUcenHbnÞúkrbs;vaRtUv)anepÞrBak;
kNþaleTAr:txagTaMgBIr.

6>60> rfynþdwkTMnijTMgn; 40.5kN GnuvtþkMlaMg
kg;dUcbgðajeTAelIr:ts<an. kMNt; (a) kMlaMg
kat;GefrFMbMputEdlekItmanenARtg;tMN C nig 6>63> KUrExST§iBlsMrab;kMlaMgenAkñúgGgát; IH
(b) m:Um:g;Bt;FMbMputenARtg;tMN C . snμt;eGay rbs; truss s<an. kMNt;kMlaMgGefrGtibrma
rfynþcrtamTisNak¾)antambeNþayG½kS ¬kMlaMgTaj b¤kMlaMgsgát;¦ EdlGacekItmanenA
kMral dUcenHbnÞúkrbs;vaRtUv)anepÞrBak;kNþal kñúgGgát;enHedaysarrfynþdwkTMnijEdlman
eTAr:txagTaMgBIr. snμt;fatMN C CatMNbgáb; TMgn; 83.25kN EdlmanbnÞúkkg;dUcbgðaj. snμt;
dUcenHvaGac Tb;Tl;)anTaMgm:Um:g;Bt; nigTaMg eGayrfynþcrtamTisNak¾)antambeNþay
kMlaMgkat;. G½kSkMral dUcenHbnÞúkrbs;vaRtUv)anepÞrBak;
kNþaleTAr:txagTaMgBIr. ehIysnμt;faGgát;
TaMgGs;RtUv)antP¢ab;KñaRtg;tMNeday gusset
plate.

6>61> kMNt;m:Um:g;GefrGtibrmaenARtg;cMnuc C
enAelIs<anEdlbNþalBIbnÞúkcl½t.
6>64> KUrExST§iBlsMrab;kMlaMgenAkñúgGgát; CD
rbs; truss s<ankñúgcMeNaT 6>63. kMNt;kMlaMg
GefrGtibrma ¬kMlaMgTaj b¤kMlaMgsgát;¦ Edl
GacekItmanenAkñúgGgát;enHedaysarrfynþdwk
6>62> rfynþmanTMgn; 21kN nigTIRbCMuTMgn;Rtg; TMnijEdlman TMgn; 83.25kN EdlmanbnÞúkkg;
G . kMNt;m:Um:g;GefrGtibrmaEdlekItmanenA dUcbgðaj. snμt; eGayrfynþcrtamTisNa k¾
cMeNaT T.Chhay -210


)antambeNþayG½kSkMral dUcenHbnÞúkrbs;va kMral dUcenHbnÞúkrbs;vaRtUv)anepÞrBak;kNþal
RtUv)anepÞrBak; kNþaleTAr:txagTaMgBIr. ehIy eTAr:txagTaMgBIr. ehIysnμt;faGgát;TaMgGs;
snμt;faGgát; TaMgGs;RtUv)antP¢ab;KñaRtg;tMN RtUv)antP¢ab;KñaRtg;tMNeday gusset plate.
eday gusset plate.
6>65> KUrExST§iBlsMrab;kMlaMgenAkñúgGgát; BC
rbs; truss s<ankñúgcMeNaT 6>63. kMNt;kMlaMg
GefrGtibrma ¬kMlaMgTaj b¤kMlaMgsgát;¦ Edl
GacekItmanenAkñúgGgát;enHedaysarrfynþdwk
TMnijEdlman TMgn; 83.25kN EdlmanbnÞúkkg; 6>68> kMNt;m:Um:g;GefrGtibrmaenARtg;cMnuc C
dUcbgðaj. snμt; eGayrfynþcrtamTisNa k¾ EdlbNþalBIbnÞúkcl½t.
)antambeNþayG½kSkMral dUcenHbnÞúkrbs;va
RtUv)anepÞrBak; kNþaleTAr:txagTaMgBIr. ehIy
snμt;faGgát; TaMgGs;RtUv)antP¢ab;KñaRtg;tMN
eday gusset plate.
6>66> kMNt;cMgay a rbs;FñwmlyedIm,IeGay
bnÞúkcl½tGacbegáItm:Um:g;Bt;GtibrmaenARtg;TMr 6>69> kMNt;kMlaMgkat;GefrGtibrmaenARtg;
esμIKñanwgm:Um:g;Bt;enAkNþalElVg. snμt; A CaTMr cMnuc C EdlbNþalBIbnÞúkcl½t ¬rUbéncMeNaT
snøak; nig B CaTMrkl;. 6>68¦.
6>70> kMNt;m:Um:g;GefrGtibrmaenARtg;cMnuc C
EdlbNþalBIbnÞúkcl½t.

6>67> KUrExST§iBlsMrab;kMlaMgenAkñúgGgát; IH
rbs; truss s<an. kMNt;kMlaMgGefrGtibrma 6>71> kMNt;kMlaMgkat;GefrGtibrmaenARtg;
¬kMlaMgTaj b¤kMlaMgsgát;¦ EdlGacekItmanenA cMnuc C EdlbNþalBIbnÞúkcl½t ¬rUbéncMeNaT
kñúgGgát;enHedaysarrfynþdwkTMnijEdlman 6>70¦.
TMgn; 25kN EdlmanbnÞúkkg;dUcbgðaj. snμt; 6>72> kMNt;kMlaMgkat; nigm:Um:g;Bt;Gtibrmadac;
eGayrfynþcrtamTisNak¾)antambeNþayG½kS xatenAkñúgFñwm AB EdlbNþalBIbnÞúk10kN .


lkçxNÐRBMEdnTamTar 0.1m ≤ x ≤ 3.9m .

6>76> bnúÞkkg;GtibrmasMrab;kg;]bkrN_sÞÚc
EdleRbIenAkñúgGKar]sSahkmμRtUv)aneGay.
]bkrN_sÞÚccrtambeNþaypøÚvrfEdlRtUv)anRT
edayTMrsamBaØenAelIssr. kMNt; (a) kMlaMg
6>73> kMNt;m:Um:g;Bt;Gtibrmadac;xatenAkñúgr:t kat;Gtibrmadac;xatenAkñúgr:tkNþal AB nig
s<anEdlbNþalBIbnÞúkrfynþdUcbgðaj. bnÞúk (b) m:Um:g;Gtibrmadac;xatenAkñúgr:t.
RtUv)andak;edaypÞal;enAelIr:t.

6>77> kMNt;m:Um:g;Bt;Gtibrmadac;xatenAkñúgr:t
6>74> kMNt;m:Um:g;Bt;Gtibrmadac;xatenAkñúgr:t s<anEdlbNþalBIbnÞúkdUcbgðaj.
s<anEdlbNþalBIbnÞúkdUcbgðaj. bnÞúkRtUv)an
dak;edaypÞal;enAelIr:t.

KMerag6>1> ]bkrN_sÞÚccl½ttambeNþaydgsÞÚc
¬ 0.1m < x < 3.4m ¦ ehIyGaceyag)an 28kN .
6>75> kMNt;m:Um:g;Bt;Gtibrmadac;xatenAkñúgr:t edayeRbIemKuNTgÁic 0.3 cUrkMNt;m:Um:g;Bt;Gti-
s<anEdlbNþalBIbnÞúkrfynþdwkTMnijdUcbgðaj. brmadac;xatenAkñúgdgsÞÚc nigkMlaMgGtibrma
EdlekItmanenAkñúgExScMNg BC .dgsÞÚcRtUv)n

cMeNaT T.Chhay -212


P¢ab;edaytMNsnøak;eTAnwgssrCBa¢aMgenAcug
xageqVgrbs;va ¬ A ¦.

KMerag6>2> s<ansMrab;GñkefμIreCIgEdlmanTMr
samBaØRtUv)aneKsg;enAkñúg]TüanTIRkugmYy
ehIyGñksikSaKNnaBIrrUb)anesñIKMeragdUcbgðaj
enAkñúgkrNI a nig b. Ggát;rbs; truss RtUv)an
sg;BIeQI. kMralpSMeLIgedaybnÞHeQIRbEvg
1.5m Edlmanm:as; 20kg / m 2 . kUdkñúgtMbn;

bgðajfabnÞúkGefrenAelIkMralEdlRtUvkarKW
5kPa CamYynwgemKuNTgÁic 0.2 . KitfakMral

RtUv)anRTedayTMrsamBaØenAelI stringer. Fñwm
kMralepÞrbnÞúkeTAtMNxageRkamrbs; truss
¬emIlrUbTI 6-23¦ enAkñúgkrNInImYy² cUrrk
Ggát;EdlrgkMlaMgTajFMCageK nigGgát;Edlrg
kMlaMgsgát;FMCageK ehIyR)ab;BImUlehtukñúgkar
eRCIserIskarKNnamYykñúgcMeNamsMeNITaMgBIr.
TMgn;rbs;Ggát; truss RtUv)anecal.


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